Asymptotic behavior in degenerate parabolic fully nonlinear equations and its application to elliptic eigenvalue problems

J. Differential Equations 254 (2013) 3259–3306

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Journal of Differential Equations www.elsevier.com/locate/jde

Asymptotic behavior in degenerate parabolic fully nonlinear equations and its application to elliptic eigenvalue problems Soojung Kim a,∗ , Ki-Ahm Lee a,b a b

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 6 February 2012 Revised 17 December 2012 Available online 5 February 2013

We study the fully nonlinear parabolic equation

MSC: 35K55 35K65

with the Dirichlet boundary condition and positive initial data in a smooth bounded domain Ω ⊂ Rn , provided that the operator F is uniformly elliptic and positively homogeneous of order one. We prove that the renormalized limit of parabolic flow u (x, t ) as t → +∞ is the corresponding positive eigenfunction which solves

Keywords: Fully nonlinear parabolic equation Large time behavior Heat equation Porous medium equation Fully nonlinear elliptic eigenvalue problem





F D 2 um − ut = 0 in Ω × (0, +∞), m  1,





F D 2 ϕ + μϕ p = 0 in Ω, 1 where 0 < p := m  1 and μ > 0 is the corresponding eigenvalue. We also show that some geometric property of the positive initial data is preserved by the parabolic flow, under the additional assumptions that Ω is convex and F is concave. As a consequence, the positive eigenfunction has such geometric property, that is, 1− p

log(ϕ ) is concave in the case p = 1, and ϕ 2 is concave for 0 < p < 1. © 2013 Elsevier Inc. All rights reserved.

Contents 1.

*

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Corresponding author. E-mail addresses: [email protected] (S. Kim), [email protected] (K.-A. Lee).

0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2013.01.015

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1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 3. Nonlinear eigenvalue problem . . . . . . . . . . . . 3.1. Case p = 1 . . . . . . . . . . . . . . . . . . . . . 3.2. Case 0 < p < 1 . . . . . . . . . . . . . . . . . . 4. Uniformly parabolic fully nonlinear equation . 4.1. Asymptotic behavior . . . . . . . . . . . . . 4.2. Log-concavity . . . . . . . . . . . . . . . . . . 5. Degenerate parabolic fully nonlinear equation 5.1. Asymptotic behavior . . . . . . . . . . . . . 5.2. Square-root concavity of the pressure . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction In this paper, we consider the asymptotic behavior of the solution u to the following fully nonlinear uniformly or degenerate parabolic equation

⎧   2 m ⎪ ⎨ F D u − ∂t u = 0 in Ω × (0, +∞), u (·, 0) > 0 in Ω, ⎪ ⎩ u=0 on ∂Ω × (0, +∞),

(1.1)

in the range of the exponents m  1, where we assume that F is uniformly elliptic and positively homogeneous of order one and Ω is a smooth bounded domain in Rn . When F is the Laplace operator, (1.1) is the well-known heat equation for m = 1, porous medium equation for m > 1, and fast diffusion equation for 0 < m < 1, respectively. For the Laplace operator, the asymptotic behavior of solutions to the degenerate or singular diffusion equations has been studied by many authors (Aronson, Berryman, Bonforte, Carrillo, Friedman, Galaktionov, Holland, Kamin, Kwong, Peletier, Toscani, Vazquez et al.). We refer to [22–24] and references therein. In this work, we first show that a renormalized flow of u (x, t ) with nonnegative initial data converges to ϕ (x) uniformly for x ∈ Ω as t → +∞, which solves the following elliptic eigenvalue problem

⎧   2 p ⎪ ⎨ F D ϕ + μϕ = 0 in Ω, ϕ>0 in Ω, ⎪ ⎩ ϕ=0 on ∂Ω,

(1.2)

1 where 0 < p := m  1 and μ > 0 is the corresponding eigenvalue depending on m. The existence and uniqueness up to a multiplicative constant of the solution to the fully nonlinear eigenvalue problem (1.2) for p = 1 were proven by Ishii and Yoshimura [14], and they also showed that the principal eigenvalue μ > 0 is unique. The simplified proof can be found at [1]. Recently, the principal eigenvalues of more general fully nonlinear operators in bounded domains have been studied by many authors and we refer to [1] and references therein. For the sub-linear case 0 < p < 1, the unique positive eigenfunction of (1.2) can be established by using a barrier argument as in Theorem 3.4. The asymptotic convergence of renormalized solutions of (1.1) to the problem (1.2) was considered in [23] for the Laplace operator in the range of 0 < p < p s , where p s is the Sobolev critical exponent 2 (p s := nn+ −2 for m  3, infinity for n = 1, 2). It has been also extended to some fully nonlinear operator F in [17] with the super-linear exponents 1 < p < pΩ, F (for some critical number pΩ, F > 1). Since a positive eigenfunction of super-linear case can be approximated by a renormalized solution of the fast diffusion equation, which will extinct in finite time, the Harnack type estimate of the solutions

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to fast diffusion equations plays an important role to analyze the asymptotic behavior near the finite extinct time. On the other hand, the solution of sub-linear eigenvalue problem will be described by renormalized solutions of (slow) diffusion equations, where the solution exists for all time. This difference allows us to have different method based on barriers for the sub-linear case. Recently, Armstrong and Trokhimtchouk in [4] concerned the long-time behavior of solutions to





F D 2 u − ∂t u = 0

in Rn × (0, +∞),

where F is a positively homogeneous, uniformly elliptic operator. They showed that rescaled solution of the Cauchy problem with nonnegative initial data converges to the positive self-similar solution Φ + (x, t ), where Φ + (·, 1) solves some elliptic eigenvalue problem. Our proof of the uniform convergence to the positive eigenfunction at Proposition 4.5 is influenced by the proof of Theorem 1.2 in [4]. Second, we show that some geometric property of the initial data is preserved by the parabolic m−1

flow, under the additional assumptions that Ω is convex and F is concave. In fact, log u and u 2 turn out to be the geometric quantities that preserve the concavity under the parabolic flow for m = 1, and m > 1, respectively. According to the uniform asymptotic convergence, the positive eigenfunction has such geometric property, and hence the super-level sets of the eigenfunction will be convex. The convexity and concavity properties of the solutions to elliptic and parabolic equations posed on a convex domain Ω ⊂ Rn have been studied by many authors since 1950. In [18], the concavity function of u

C (x, y ; u ) =

1 2

 x+ y u (x) + u ( y ) − u for (x, y ) ∈ Ω × Ω 2

was introduced by Korevaar to study the convexity of super-level sets of solutions to elliptic and parabolic equations, and of course, the concavity of u is equivalent to the nonpositivity of C (x, y ; u ). Korevaar used a maximum principle and some boundary value conditions to show the log-concavity of the positive eigenfunction of (1.2) for F = , and p = 1, simplifying the proof of Brascamp and Lieb [5]. Korevaar’s idea was extended to the case 0 < p < 1 with the Laplace operator by Kawohl [15] (see also [9]). Lin [20] and Gladiali and Grossi [12] concerned the convexity of the super-level sets of the energy minimizer and the energy minimizing sequence, respectively, for the case 0 < p < p s with the Laplace operator. Alvarez, Lasry, and Lions [2] developed a new approach to the convexity problem of viscosity solutions to a nonlinear degenerate elliptic equation





F x, u (x), ∇ u (x), D 2 u (x) = 0 in Ω, under the assumption that Ω is a convex domain in Rn and that

  (x, r , A ) → F x, r , p , A −1 is concave for (x, r , p , A ) ∈ Ω × R × Rn × Sn+×n , where Sn+×n denotes the set of n × n symmetric nonnegative matrices. In [21], the authors found the necessary and sufficient conditions for the propagation of convexity of solutions to linear and fully nonlinear parabolic equations in Rn × R+ . 1.1. Let Sn×n denote the set of n × n symmetric matrices and M denote the norm of a matrix M ∈ Sn×n , which is defined as the maximum absolute value among eigenvalues of M. For 0 < λ  Λ (called ellipticity constants), Pucci’s extremal operators, that play a crucial role in the study of fully nonlinear elliptic equations, are defined as follows: for M ∈ Sn×n , + M+ λ,Λ ( M ) := M ( M ) = sup tr ( AM ), A ∈Aλ,Λ

− M− tr( AM ), λ,Λ ( M ) := M ( M ) = A ∈inf Aλ,Λ

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where Aλ,Λ consists of the symmetric matrices whose eigenvalues lie in [λ, Λ]. We note that when λ = Λ = 1, Pucci’s extremal operators M± simply coincide with the Laplace operator. In this article, the fully nonlinear operator F : Sn×n → R will be commonly assumed to satisfy the following basic hypotheses:

( F 1) F is a uniformly elliptic operator; for all M , N ∈ Sn×n ,

M− ( M − N )  F ( M ) − F ( N )  M+ ( M − N ), ( F 2) F is positively homogeneous of order one; for all t  0 and M ∈ Sn×n , F (t M ) = t F ( M ),

( F 3) F is concave. We may extend F on Rn by defining F ( A ) := F ( A +2A ) for a nonsymmetric matrix A. One of nontrivial examples of the operator satisfying ( F 1), ( F 2), and ( F 3) is Pucci’s extremal operator M− . We remark that the concavity of F , ( F 3), will be required when we show geometric property of the solutions to the elliptic and parabolic equations. Throughout this paper, we always assume that Ω is a bounded domain with a smooth boundary in Rn . We recall viscosity solutions, which are the proper notion of the weak solutions for the fully nonlinear elliptic and parabolic equations in nondivergence form. Let F be a uniformly elliptic operator, and f , and u be continuous functions defined on a bounded domain Ω ⊂ Rn . A continuous function u is said to be a viscosity sub-solution (respectively, viscosity super-solution) of T

2





F D2u = f

in Ω

when the following condition holds: for any x0 ∈ Ω and φ ∈ C 2 (Ω) such that u − φ has a local maximum at x0 , we have





F D 2 φ(x0 )  f (x0 ) (respectively, if u − φ has a local minimum at x0 , we have F ( D 2 φ(x0 ))  f (x0 )). We say that u is a viscosity solution of F ( D 2 u ) = f in Ω when it is both a viscosity sub-solution and super-solution. The existence of viscosity solutions can be established by Perron’s method via the comparison principle. Viscosity solutions of parabolic equations are defined analogously. We refer the details and regularity theory of the viscosity solutions of the fully nonlinear elliptic and parabolic equations to [7,6,25,26] and references therein. 1.2.

When the operator is fully nonlinear, there are several issues to be discussed.

(i) Parabolic approach relies on the convergence of the parabolic flow u (x, t ) to eigenfunction ϕ (x), after normalization. For parabolic operators of divergence type, some key steps for asymptotic analysis are based on integration by parts, for example, the existence of monotone integral quantities in [24], which cannot be applicable to the operators in nondivergence form. On the other hand, asymptotic analysis for the operators of nondivergent form can be achieved in a couple of steps. Especially, it is crucial to find an exact decay rate of u (x, t ), as t tends to +∞, which gives us the right renormalization of u so that the renormalized limit will be the positive eigenfunction. In fact, the exact decay rate is related to the principal eigenvalue μ > 0 when m = 1. In Proposition 4.5, we show the existence of the unique limit of renormalized function e μt u (x, t ), where the limit is the positive eigenfunction ϕ (x). When 1 < m = 1p < ∞, we

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prove Aronson–Bénilan type estimate, Lemma 5.7, Corollary 5.8 for degenerate parabolic operators, which gives us almost monotonicity of u (x, t ). For the case, the uniqueness of the limit of m the renormalized solutions t m−1 um (x, t ) is proven at Proposition 5.2 and Corollary 5.3. (ii) When we perform geometric computation, we first investigate the smooth solutions for smooth operators, and then use approximation to obtain the result for general operators. But, we cannot approximate the operator F satisfying ( F 1), ( F 2), and ( F 3), by a smooth operator F ε satisfying the same assumptions as F . If a differentiable operator F satisfies ( F 2), the operator F has to be linear, and F will be the Laplacian under the additional assumption ( F 1) after simple modification. Instead, we have Lemma 4.7 to approximate F by a smooth one satisfying ( F 1), ( F 3), and (4.11), which replaces the assumption ( F 2). So sophisticated geometric computations are required to investigate geometric quantities, which satisfy maximum principle in Lemmas 4.9, 5.15. 1.3. This paper is organized into four parts as follows. In Section 2, we summarize basic facts and estimates about fully nonlinear uniformly elliptic and parabolic equations. In Section 3, we concern the existence and uniqueness of the fully nonlinear elliptic eigenvalue problem (1.2). In Section 4, we deal with the fully nonlinear uniformly parabolic equations. We show the uniform convergence of the renormalized parabolic flow to the positive eigenfunction in Proposition 4.5, and the propagation of log-concavity of the solution u (x, t ) in Lemma 4.9. Consequently, log-concavity of the positive eigenfunction can be proven. In Section 5, we concern the degenerate parabolic equations in the range of the exponents m > 1. We show the asymptotic convergence to the positive eigenfunction in Proposition 5.2. It is also proved that the pressure um−1 of the solution preserves square-root concavity in Lemma 5.15 under some flow, and hence the power concavity of the positive eigenfunction is established. Notations. Let us summarize the notations and definitions that will be used.

• We denote by ∇ u or Du the spatial gradient of a function u (x, t ), and by D 2 u the Hessian matrix. D e f denotes the directional derivative in the direction e ∈ Rn . • The expressions D 2 u  0, D 2 u  0 are understood in the usual sense of quadratic forms, for instance, D 2 u  0 means that D 2 u is positive-semidefinite. • In order to avoid confusion between coordinates and partial derivatives, we will use the standard subindex notation to denote the former, while partial derivatives will be denoted in the form f ,α ∂f

for ∂ x = ∂α f . In general, f ,α = ∇eα f for a unit vector e α ∈ Rn with a parameter α ∂2 f partial derivatives will be denoted in the form f ,α β for ∂ x ∂ x α β

α . Second order

= e β D f e α = ∂α β f . If the comput

2

tation is invariant under the rotation, we may assume that α = 1, . . . , n, and that {e 1 , . . . , en } is an orthonormal basis. This notation is usual in some parts of the physics literature. But we will write f ν and f τ for the normal and tangential derivatives since no confusion is expected. 2. Preliminaries For the reader’s convenience, we summarize basic facts and estimates for fully nonlinear elliptic equations





F D2u = f

in Ω,

[7,6], and for fully nonlinear parabolic equations





F D 2 u − ∂t u = f

in Ω × (0, T ],

[7,19,25,26], where we assume that F satisfies ( F 1), and Ω is a smooth bounded domain in Rn for this section.

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(1) The existence and uniqueness of the viscosity solution, comparison principle between super- and subsolutions, and the maximum principle for the elliptic or parabolic Dirichlet problem, and their references can be found in [7,6,25]. (2) The strong maximum principle holds if f = 0 in the elliptic equations (see Proposition 4.9 and Corollary 3.7 in [6]). A similar argument with Corollary 3.21 and Corollary 4.14 in [25] will give the strong maximum principle for the parabolic equations (see also [19]). The strong maximum principle for the elliptic or parabolic equation says that whenever a sub-solution u touches a super-solution v from below at an interior point of Ω or Ω × (0, T ], then u ≡ v in the domain. (3) [Local regularity] We refer the regularity theory for the elliptic equations to [13,6] and the parabolic case to [19,25,26]. In [6], one can find the following: (a) interior Hölder regularity for f ∈ L n (Ω) at Proposition 4.10, (b) interior C 1,α -regularity (0 < α < 1) for a Hölder continuous function f at Theorem 8.3 and Corollary 5.7, (c) interior C 1,1 -regularity for concave (or convex) equations F ( D 2 u ) = 0 in the proof of Theorem 6.6, (d) interior W 2, p -regularity (n < p < ∞) for F ( D 2 u ) = f with a concave (or convex) operator F , and f ∈ L p (Ω) at Theorem 7.1, (e) interior C 2,α -regularity (0 < α < 1) for F ( D 2 u ) = f with a concave (or convex) operator F , and a Hölder continuous function f at Theorem 8.1, (f) interior C ∞ -regularity for F ( D 2 u ) = f with a smooth, and concave (or convex) operator F , and a smooth function f . When a concave (or convex) operator F , and f are analytic, u will be analytic following Theorem 10 in [11, Section 2.2]. (4) [Global regularity] We also refer the global Schauder theory:

  u C k+2,α (Ω)  C u L ∞ (Ω) + f C k+α (Ω) , to Theorem 9.5, Proposition 9.8 in [6], where we also assume that a concave (or convex) operator F is C k , and the boundary data u |∂Ω is C k+3 . Therefore, viscosity solutions will be classical. Similar results hold for parabolic equations [19,25,26]. (5) [Harnack inequality] The Harnack inequality for nonnegative solutions to elliptic equations is the following (see [6, Theorem 4.8]): for a nonnegative solution u of F ( D 2 u ) = f in B 2 (0), we have

sup u  C

B 1 ( 0)



inf u + f Ln ( B 2 (0))

B 1 ( 0)

for a uniform constant C > 0. The parabolic Harnack inequality can be found in [25]. 3. Nonlinear eigenvalue problem Throughout this section, we assume that Ω is a smooth and bounded domain in Rn . In this section, we study the existence, uniqueness and some useful properties of solutions to the fully nonlinear elliptic eigenvalue problem

⎧   2 p ⎪ ⎨ F D φ(x) = −μφ (x) in Ω, φ>0 in Ω, ⎪ ⎩ φ=0 on ∂Ω,

(NLEV)

for 0 < p  1, under suitable condition on a uniformly elliptic operator F . The operator F can be assumed additionally to satisfy ( F 2) or ( F 3). For the existence theorem of problem (NLEV), we will assume that F satisfies ( F 1) and ( F 2).

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3.1. Case p = 1 First, let us introduce the existence and uniqueness up to a multiplicative constant to problem (EV) that was proven by Ishii and Yoshimura [14]. The simplified proof can be found at [1]. Theorem 3.1. (See [14].) Suppose that F satisfies ( F 1) and ( F 2). Then there exist ϕ ∈ C 1,α (Ω) (0 < α < 1), and μ > 0 such that ϕ > 0 in Ω and ϕ satisfies



  − F D 2 ϕ (x) = μϕ (x) in Ω,

ϕ (x) = 0

on ∂Ω.

(EV)

Moreover, μ is unique in the sense that if ρ is another eigenvalue of F in Ω associated with a nonnegative eigenfunction, then μ = ρ ; and is simple in the sense that if ψ in C 0 (Ω) is a solution of (EV) with ψ in place of ϕ , then ψ is a constant multiple of ϕ . Now we state Hopf ’s Lemma that will be used frequently when we compare a solution with a barrier. Theorem 3.2 (Hopf ’s Lemma). (i) Let u ∈ C (Ω) be a nonzero super-solution of

  M− D 2 u  0 in Ω. Then, for xo ∈ ∂Ω satisfying u (x) > u (xo ) for all x ∈ Ω , we have

lim inf

u (x) − u (xo )

Ωx→xo

|x − xo |

> 0.

In particular, if the outer normal derivative of u at xo exists, then

∂u (xo ) < 0, ∂ν where ν is the outer normal vector to ∂Ω at xo . (ii) Let u ∈ C (Ω × [0, T ]) be a nonzero super-solution of

  M− D 2 u − ut  0 in Ω × (0, T ]. For (xo , to ) ∈ ∂Ω × (0, T ] satisfying u (x, t ) > u (xo , to ) for all x ∈ Ω × (0, t o ], we have

lim inf

Ωx→xo , t ↑to

u (x, t ) − u (xo , to )  > 0. |x − xo |2 + (to − t )

We refer to Lemma 3.4 at [13] for uniformly elliptic linear equation and Appendix at [1] for uniformly elliptic fully nonlinear equation. In fact, Hopf ’s Lemma for uniformly elliptic equation follows from the comparison between a super-solution and a barrier of the form R −α − |x|−α for large α > 0 and a small R > 0. We refer to Lemma 2.6 [19] for uniformly parabolic equation.

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3.2. Case 0 < p < 1 In this case, we consider the following elliptic equation

⎧  2 m  1 ⎪ ⎪ ⎨ − F D f (x) = m − 1 f (x) in Ω, m > 1, f =0 on ∂Ω, ⎪ ⎪ ⎩ f >0 in Ω,

(3.3)

which is the asymptotic profile, after appropriate normalization (Proposition 5.2), of solutions to the following parabolic equation

⎧  2 m  ⎪ ⎨ ut (x, t ) − F D u (x, t ) = 0 in Ω × (0, +∞), m > 1, u (x, 0) = u o (x) in Ω, ⎪ ⎩ u (x, t ) = 0 on ∂Ω × (0, +∞).

(3.4)

For (3.4), we require initial data u o to have nontrivial bounded gradient of um o on ∂Ω , i.e.,

um o ∈ Cb (Ω), where

  Cb (Ω) := h ∈ C 0 (Ω)  co dist(x, ∂Ω)  h(x)  C o dist(x, ∂Ω) for 0 < co  C o < +∞ . So, if f is a solution of (3.3), then φ := f m is a solution of (NLEV) associated with the exponent 1 1 p=m and the eigenvalue μ = m− . 1

1 For the sub-linear case, 0 < p = m < 1, we can generalize the comparison principle as follows and we refer to Section 2 [3] for Laplace operator. We also found that Comparison Theorem 3.3 at [1] dealt with viscosity sub- and super-solutions of general elliptic equation.

Lemma 3.3 (Comparison principle). Suppose that F satisfies ( F 1), F (0) = 0 and either ( F 2) or ( F 3). Let v and w be in C 2 (Ω) ∩ C 1 (Ω) such that v ≡ 0, v , w  0 in Ω and





F D2 v +

1 m−1

1





v m  0  F D2 w +

1 m−1

1

wm

in Ω.

If v  0  w on ∂Ω , then v  w in Ω . 1

1 v m < 0 in Ω . Then Proof. (i) First, we assume that v is a strict super-solution, i.e., F ( D 2 v ) + m− 1 we will show that v > w in Ω . On the contrary, suppose that v  w for some point in Ω . Since a nonzero and nonnegative function v satisfies

    M− D 2 v  F D 2 v < −

1 m−1

1

v m  0 in Ω,

we have that v > 0 in Ω and |∇ v | > δ0 > 0 on ∂Ω ∩ { v = 0} for some δ0 > 0 from the strong minimum principle and Hopf ’s Lemma 3.2. This implies that there is a small ε > 0 such that v  ε w since w ∈ C 1 (Ω) and w = 0 on ∂Ω . Define

t ∗ := sup{t > 0 | v  t w in Ω}.

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Then 0 < ε  t ∗  1 from the assumption that v  w for some point in Ω since v is positive in Ω . Now we set z := v − t ∗ w and then a nonnegative function z vanishes at some point in Ω and satisfies

      M− D 2 z  F D 2 v − F D 2 t ∗ w < 



1

t∗ w

m−1

 m1

1 m

−v





1 m−1

1

1

t∗ w m − v m



 0 in Ω,

where we use the hypotheses on the operator F in the first two inequalities. Since M− ( D 2 z) < 0 in Ω , we have that z ≡ 0, so the strong minimum principle and Hopf ’s Lemma imply that z > 0 in Ω and |∇ z| > δ1 > 0 on ∂Ω ∩ { z = 0} for some δ1 > 0. Then we can choose ε0 > 0 small so that z  ε0 w in Ω , since w = 0 on ∂Ω and w ∈ C 1 (Ω). It is a contradiction to the definition of t ∗ . Therefore, we conclude that v > w in Ω . 1 1 (ii) Now we assume that v is a super-solution, i.e., F ( D 2 v ) + m− v m  0 in Ω and we approxi1 mate v by strict super-solutions. We note that v > 0 in Ω from the strong minimum principle. Let v ε := (1 + ε ) v for 0 < ε < 1. Then v ε satisfies v ε > v and





F D2 vε +

1

 ε  m1   (1 + ε ) m 1 v  (1 + ε ) F D 2 v + vm m−1 m−1 1  1 1  v m (1 + ε ) m − (1 + ε ) < 0 in Ω. m−1 1

From (i), we get v ε > w in Ω . Letting

ε → 0, it follows that v  w in Ω . 2

Theorem 3.4 (Existence and uniqueness). Suppose F satisfies ( F 1) and ( F 2). Let 0 < p := ear eigenvalue problem (NLEV) has a unique solution φ ∈ C

⎧  2  ⎪ ⎪ ⎨ −F D φ = φ=0 ⎪ ⎪ ⎩ φ>0

0,1

1 m−1

(Ω) ∩ C

1, α

1 m

< 1. The nonlin(Ω) (0 < α < 1), namely, φ satisfies

φ p in Ω, (NLEV)

on ∂Ω, in Ω.

Proof. (i) First, the uniqueness of the solution follows from the comparison principle. To prove the existence, we use Perron’s method via comparison principle so we establish positive super- and subsolutions with zero boundary data. Let h be the solution of

⎧  2  ⎨ F D h = −1 in Ω, on ∂Ω, ⎩h = 0 h>0

−1

1

If we select t > 0 satisfying t 1− m h L ∞m(Ω) =





in Ω.

1 , m−1

1

then we have

1

1

F D 2 (th) = −t = −t 1− m h− m (th) m  −

1 m−1

i.e., h+ := th is a super-solution. Now, let choosing s > 0 such that

1

(th) m

in Ω,

ϕ be the positive eigenfunction of (EV) at Theorem 3.1. By 1 1 μ(s ϕ L ∞ (Ω) )1− m = m− , we have 1 



F D 2 (sϕ )  −

1 m−1

1

(sϕ ) m

in Ω.

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Thus, h− := sϕ is a sub-solution. So we have constructed a super-solution h+ and a sub-solution h− . Comparison principle implies that h−  h+ and then there is a viscosity solution φ to (NLEV) such that h−  φ  h+ from Perron’s method [7]. (ii) Now we show that φ ∈ C 0,1 (Ω) ∩ C 1,α (Ω). First, φ belongs to L ∞ (Ω) from comparison since + h ∈ L ∞ (Ω) by ABP-estimates [6]. Then φ is of C 1,α (Ω) from the regularity theory of uniformly elliptic equations [6] since φ p ∈ L ∞ (Ω). To show Lipschitz regularity of φ up to the boundary, we recall that

co dist(x, ∂Ω)  h−  φ  h+  C o dist(x, ∂Ω)

(3.5)

for some 0 < co  C o < +∞ from  Hopf ’s Lemma for h− and C 0,1 (Ω)-regularity of h+ [6]. Let δ > 0 be a fixed constant such that x∈Ω(−ε) B ε (x) ⊂ Ω for any 0 < ε  δ , where Ω(−ε) := {x ∈ Ω: dist(x, ∂Ω) > ε }. For xo ∈ Ω \ Ω(−δ) , set

2ε := dist(xo , ∂Ω)  δ. Now we scale the function φ in the following way:

1

φε (x) := φ(xo + ε x) in B 1 (0).

ε

Then (3.5) implies that 0 < co  φε (x)  3C o in B 1 (0) and φε satisfies





F D 2 φε = −

ε 1+ p m−1

p

φε

in Ω

p

with φε L ∞ ( B 1 (0))  (3C o ) p . From the regularity theory [6], we have

     D φ(xo ) =  D φε (0)  C˜ for a uniform constant C˜ > 0, where C˜ depends only on C o , p, λ, Λ and n. Therefore, we have | D φ(xo )|  C˜ for any xo ∈ Ω \ Ω(−δ) and then we conclude that

D φ L ∞ (Ω)  C˜ + D φ L ∞ (Ω (−δ) ) < +∞ from the interior C 1,α -estimates.

2

Remark 3.5. (i) From Hopf ’s Lemma, the eigenfunction φ in Theorem 3.4 has nontrivial bounded gradient on the boundary, that is, inf∂Ω |∇φ| > δo > 0 for some δo > 0. So, φ belongs to Cb (Ω) from Lipschitz regularity. (ii) Under the extra condition that F is of C 1 , φ in Theorem 3.4 is of C ∞ (Ω) since C 1 positively homogeneous operator F of order one is a constant linear operator which can be transformed to Laplacian. We finish this section with the existence and uniqueness of solutions to problem (3.4). For the purpose, we use Perron’s method and then make use of the positive eigenfunction φ in order to construct a separable solution to (3.4). By setting v := um , we consider the following parabolic equation:

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⎧    1 2 ⎪ in Q = Ω × (0, +∞), m > 1, ⎨F D v = vm t m v (x, 0) = v o (x) = u o ∈ Cb (Ω), ⎪ ⎩ v =0 on x ∈ ∂Ω × (0, +∞).

(3.6)

First, we state the comparison principle of (3.6). A similar argument as Lemma 3.3 will give us the following comparison and the proof for the case m > 1 is the same as the case mΩ, F < m < 1 at [17] (see also [3] for the porous medium equation). Lemma 3.6 (Comparison principle). Suppose F satisfies ( F 1), F (0) = 0 and either ( F 2) or ( F 3). Let v , w ∈ C 2,1 ( Q T ) ∩ C 0 ( Q T ) satisfy that v , w > 0 in Q T ,







1

F D2 v − v m



   1  0  F D 2 w − w m t in Q T ,

t

v  w on Ω × {0}, and v  0  w on ∂Ω × (0, T ], where Q T := Ω × (0, T ]. Then we have that v  w in Q T . Theorem 3.7 (Existence and uniqueness). Suppose that F satisfies ( F 1), F (0) = 0 and either ( F 2) or ( F 3). Let m > 1. When um o is in Cb (Ω), there exists a unique solution of porous medium type equation (3.4). Moreover, the solution u is positive in Ω × (0, +∞). Proof. (i) For the existence, we use Perron’s method through comparison principle so it suffices to construct a super-solution and a sub-solution. Let φ + be the positive eigenfunction associated with Pucci’s operator M+ in place of F in Theorem 3.4. We recall that Pucci’s operators M± satisfy ( F 1) 1

and ( F 2). Then it is easy to check that (φ + )1/m (x)(k + t )− m−1 (for any k > 0) solves



w t = M+ D 2 w m



in Ω × (0, +∞),

with zero lateral boundary data. We choose K > 0 small so that m

− m −1 + 0 < um o (x)  φ (x) K 1

− m−1 + + + m since um , u + solves o ∈ Cb (Ω) and inf∂Ω |∇φ | > 0. Setting u (x, t ) := (φ ) (x)( K + t ) 1

⎧  2 m + ⎪ in Ω × (0, +∞), ⎨ wt − M D w = 0 + w (x, 0) = u (x, 0)  u o (x) in Ω, ⎪ ⎩ w (x, t ) = 0 on ∂Ω × (0, +∞). Since F  M+ , u + satisfies ut+ − F ( D 2 (u + )m )  0 in Ω × (0, +∞). Thus u + is a super-solution of (3.4). We note that 0 is a trivial sub-solution. Therefore Perron’s method together with comparison principle [7], will imply that there exists a unique solution u of (3.4) such that 1

1

0  u (x, t )  φ + m (x)( K + t )− m−1

in Ω × (0, +∞).

The uniqueness follows from comparison principle. (ii) We show that u > 0 in Ω × (0, +∞). Let Ω  be a smooth bounded domain such that Ω   Ω . From Theorem 3.4, let φ − be the unique solution of

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⎧  2  − ⎪ ⎪ ⎨ −M D φ =

1 m−1

φ=0 ⎪ ⎪ ⎩ φ>0

φ p in Ω  , on ∂Ω  , in Ω  . m

− m −1 −  Since um  um o is positive in Ω , we find K > 0 large so that 0 < φ (x) K o (x) on Ω . Setting 1

u − (x, t ) := (φ − )1/m (x)( K + t )− m−1 , u − solves

⎧  2 m − ⎪ in Ω  × (0, +∞), ⎨ wt − M D w = 0 − w (x, 0) = u (x, 0)  u o (x) in Ω  , ⎪ ⎩ w (x, t ) = 0 on ∂Ω  × (0, +∞). Since M−  F , u − satisfies ut− − F ( D 2 (u − )m )  0 in Ω  × (0, +∞). Thus the comparison principle in Ω  × (0, +∞) gives that u (x, t )  u − (x, t ) > 0 in Ω  × (0, +∞). By taking Ω  arbitrary, we conclude that u > 0 in Ω × (0, +∞). 2 4. Uniformly parabolic fully nonlinear equation In this section, we study fully nonlinear uniformly parabolic equation

⎧  2  in Ω × (0, +∞), ⎪ ⎨ F D u − ∂t u = 0 0 u (·, 0) = u o ∈ C (Ω), ⎪ ⎩ u=0 on ∂Ω × (0, +∞).

(4.7)

In the entire section, we assume that Ω is a smooth bounded domain of Rn and F satisfies ( F 1). We also assume that initial data u o ∈ C 0 (Ω) is nonnegative in Ω . 4.1. Asymptotic behavior In this subsection, we analyze the asymptotic behavior of the solution u of (4.7) when time goes infinity. First, we find the exact decay rate of u comparing with barriers which are separable solutions of the form ϕ (x)e −μt , where ϕ (x) is the positive eigenfunction and μ > 0 is the principal eigenvalue in Theorem 3.1. Lemma 4.1. Suppose F satisfies ( F 1) and ( F 2). Let u be the solution of (4.7) and let ϕ be the positive eigenfunction with the eigenvalue μ > 0 in Theorem 3.1. For u o ∈ Cb (Ω), there are 0 < C 1 < C 2 < +∞ such that

C 1 ϕ (x)e −μt < u (x, t ) < C 2 ϕ (x)e −μt ,

∀(x, t ) ∈ Ω × (0, +∞).

Proof. By Hopf ’s Lemma and Lipschitz regularity of ϕ , we have 0 < |∇ ϕ | < +∞ on ∂Ω . So we can choose C 2 > C 1 > 0 such that C 1 ϕ (x) < u o (x) < C 2 ϕ (x) in Ω since u o ∈ Cb (Ω). Since C ϕ (x)e −μt (for any C > 0) is a solution of (4.7) with initial data C ϕ (x), the comparison principle gives us the result. 2 Lemma 4.2. Suppose that F satisfies ( F 1) and ( F 2). Let u be the solution of (4.7) and let ϕ be the positive eigenfunction associated with the eigenvalue μ > 0 in Theorem 3.1. For a nonnegative and nonzero u o ∈ C 0 (Ω), there exist T o > 0 and 0 < C 1 < C 2 < +∞ such that

C 1 ϕ (x)e −μ T o < u (x, T o ) < C 2 ϕ (x)e −μ T o ,

∀x ∈ Ω.

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Moreover, we have

C 1 ϕ (x)e −μt < u (x, t ) < C 2 ϕ (x)e −μt ,

∀(x, t ) ∈ Ω × [ T o , +∞).

(4.8)

Proof. (i) First, we construct a sub-solution of F ( D 2 w ) − w t = 0, whose support expands in time. Define g (x, t ) = (r , 0, . . . , 0),

1 tβ

2

exp(−α rt ) for

α=

1 , 4λ

β=

Λn and r 2λ

= |x|. Then it is easy to see that at the point

∂i j g = 0 if i = j ,  g ∂11 g = 2α 2 2α r 2 − t , t

∂ii g = −2α

g t

and

if i > 1.

Since M− is rotationally symmetric, we can check for r 2 < 2tα

  g M− D 2 g − gt = Λ∂11 g + (n − 1)Λ∂22 g − gt = 2 t (β − 2α Λn) + α r 2 (4Λα − 1)  0 t

and for r 2  2tα

    g M− D 2 g − gt = 2 t β − 2α λ + (n − 1)Λ + α r 2 (4λα − 1)  0. t

This implies that g satisfies F ( D 2 u ) − ut  M− ( D 2 u ) − ut  0. Now we define for any xo ∈ Ω ,

 h(x, t ) := max co



|x − x0 |2 exp −α β (t + τo ) t + τo 1



 − δo , 0 ,

where positive constants co , τo , and δo will be chosen later. Then h is also a sub-solution of F ( D 2 w ) − w t = 0 as long as supp h(·, t ) ⊂ Ω . Select a point xo and η > 0 such that u o (xo ) > 0, u o > 0 on B η (xo ) and B 2η (xo ) ⊂ Ω . We recall  that u o ∈ C 0 (Ω) is nonzero and nonnegative. We assume that η > 0 satisfies x∈Ω(−2η) B 2η (x) ⊂ Ω for

Ω(−2η) := {x ∈ Ω : dist(x, ∂Ω) > 2η}. We set m0 := min B η (xo ) u o > 0. By choosing co , τo and δo such that

η2 = 4Λnτo (> 2Λnτo ),



co

η2

exp −α β τo

τo

 = δo and

co

τoβ

− δo = m0 ,

we can show that supp h(·, 0) ⊂ B η (xo ) and h(x, 0)  mo in B η (xo ), and that the support of h(x, t ) is

1 2 increasing for 0 < t  t 0 := 1e ( cδo )1/β − τo = 4e− Λλ η . We also have that o

supp h(·, t 0 ) = B √ e η (xo ) at t 0 = 2

e−1 4Λλ

η2 .

Comparison principle implies that h(x, t )  u (x, t ) in Ω × (0, t 0 ] and hence u (x, t 0 )  h(x, t 0 ) > 0 in 1 2 B √ e η (xo ) at t 0 = 4e− Λλ η > 0. So far, we have proved that if u (·, 0) > 0 on B η (xo ), then 2

e−1 2 u (·, t 0 ) > 0 on B √ e η (xo ) at t 0 = η > 0. 2 4Λλ By setting 1 + 2ε :=



e , 2

we also have that u (·, t 0 ) > 0 in B (1+ε)η (xo ).

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(ii) Now, we apply the above argument repeatedly to show that u (·, T o ) > 0 in Ω for some T o > 0. For any y ∈ Ω(−(1+2ε)η) := {x ∈ Ω : dist(x, ∂Ω) > (1 + 2ε )η}, we have a chain of uniform number of balls { B η (xk )}kN=1 such that x1 = xo , ∂ B εη (x N ) = y, and |xk − xk+1 |  εη for k = 1, . . . , N − 1. The number N ∈ N of balls is bounded by a uniform constant depending on ε , η , n and Ω . Applying (i) N times with barriers h (after suitable translation in space variables at each step), we deduce that there exists a time t 1 > 0 such that u (·, t 1 ) > 0 on Ω(−2ε) . Indeed, once u (x, t k ) > 0 in some ball B (1+ε)η (xk ) at t = t k , then u > 0 in B (1+ε)η (xk ) × [t k , +∞), by comparison with a separable solution δk ϕ1 (x)e −μ1 t for small δk > 0, where ϕ1 is the positive eigenfunction in B (1+ε+εk )η (xk ) (for some small εk > 0) associated with μ1 > 0. Therefore, we conclude that there is a time T o > 0 such that

u (x, T o ) > 0 in Ω

and

  ∇ u ( y , T o ) > 0,

∀ y ∈ ∂Ω

by applying (i) again. The second inequality comes from the nontrivial gradient property of the barrier h. (iii) We choose C 1 > 0 small such that C 1 ϕ (x)e −μ T o < u (x, T o ) in Ω since |∇ u ( y , T o )| > 0 for any y ∈ ∂Ω . Since u is C 1+γ (Ω × [ T o , T o + 1]), there is C 2 > 0 such that

C 1 ϕ (x)e −μ T o < u (x, T o ) < C 2 ϕ (x)e −μ T o

in Ω.

2

Therefore, the comparison principle implies that (4.8).

Under the assumption that F satisfies ( F 1) and ( F 2), we refine the asymptotic behavior of solutions to (4.7). Let u be the solution of (4.7) and μ be the principal eigenvalue in Theorem 3.1. Define the normalized function

v (x, t ) := e μt u (x, t ).

(4.9)

Then, v (x, t ) satisfies





vt = F D 2 v + μv

in Ω × (0, +∞).

From Lemma 4.2, we deduce the following corollary. Corollary 4.3. Under the same assumption of Lemma 4.2, v (x, t ) = e μt u (x, t ) has the following estimates:

C 1 ϕ (x) < v (x, t ) < C 2 ϕ (x)

   v (x, t )

L ∞ (Ω×[ T o ,+∞))



in Ω × [ T o , +∞), C2   C1



v (x, T o ) L ∞ (Ω) ,

where 0 < C 1 < C 2 < +∞ and T o > 0 are in Lemma 4.2. Proof. From (4.8), we have

C 1 ϕ (x) < v (x, t ) < C 2 ϕ (x)  and hence the result follows.

C 2 μT o C2 e u (x, T o ) = v (x, T o ) in Ω × [ T o , +∞), C1 C1

2

Before studying fine asymptotic behavior of solutions to (4.7), let us summarize the regularity theory of uniformly parabolic equations. We refer to [13,6,19,25,26].

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Theorem 4.4 (Global regularity for m = 1). Suppose that F satisfies ( F 1). (i) Let u solve F ( D 2 u ) − ut = 0 in Ω × (0, +∞). For each compact subset K of Ω × (0, +∞): (a) u is of C 1,β ( K ) for some 0 < β < 1. (b) If F is concave, u is of C 1,1 ( K ). (c) If u ∈ C 1,1 ( K ) and F is concave, then u is of C 2,β ( K ) for some 0 < β < 1. (d) If u ∈ C 2,β ( K ) and F ∈ C ∞ , then u is of C ∞ ( K ). (ii) Let v be a bounded solution of F ( D 2 v ) + μ v − v t = 0 in Ω × (0, T ]. Then, (a), (b), (c) and (d) for v also hold. We note that C β means that C β in x and C β/2 in t in the parabolic setting. Now, we show that the normalized parabolic flow v (x, t ) = e μt u (x, t ) converges uniformly to the unique limit as t → +∞, which is the positive eigenfunction in Theorem 3.1. In order to obtain the uniform convergence to the positive eigenfunction, we use the approach presented at [4]. Armstrong and Trokhimtchouk in [4] studied the long-time behavior of solutions to the uniformly parabolic equation in Rn × R+ . Proposition 4.5. Suppose F satisfies ( F 1) and ( F 2). Let u be the solution of (4.7) with a nonzero nonnegative initial data u o ∈ C 0 (Ω) and let ϕ be the positive eigenfunction associated with the eigenvalue μ > 0 in Theorem 3.1. Define v (x, t ) := e μt u (x, t ). Then, there exists a unique constant γ ∗ > 0 depending on u o such that

   v (x, t ) − γ ∗ ϕ (x)

C x0 (Ω)

→ 0 as t → +∞.

Proof. We recall that v is bounded from Corollary 4.3 and then





sup v (·, · + s)C α (Ω×[0,+∞)) < +∞ for 0 < α < 1, s1

from the uniform Hölder regularity (see Theorem 4.23 [25]). For a given sequence {sn } such that sn → +∞ as n → +∞, we find a subsequence {snk } and a function w ∈ C α (Ω × [0, +∞)) such that

v (x, t + snk ) → w (x, t )

locally uniformly in Ω × [0, +∞) as nk → +∞,

according to Arzelà–Ascoli Theorem. Then a limit w satisfies F ( D 2 w ) + μ w − w t = 0 in Ω × (0, +∞). Now, let A be the set of all sequential limits of { v (·, · + s)}s T o , where T o > 0 is given in Corollary 4.3. Then any w ∈ A satisfies that





F D 2 w + μ w − w t = 0 in Ω × (0, +∞) and

C 1 ϕ (x)  w (x, t )  C 2 ϕ (x)

in Ω × (0, +∞)

from Corollary 4.3, where the constants 0 < C 1 < C 2 < +∞ are in Corollary 4.3. We define





γ ∗ := inf γ > 0: ∃ w ∈ A such that w  γ ϕ in Ω × (0, +∞) . We note that 0 < C 1  γ ∗  C 2 < +∞. We are going to prove that A = {γ ∗ ϕ }. ˜ ∈ A such that w ˜  (γ ∗ + ε )ϕ First, we show that w  γ ∗ ϕ for any w ∈ A. Fix ε > 0. There exists w ˜ by the definition of γ ∗ and then we have a sequence of functions, { v n := v (·, · + sn )}, converging to w

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˜ (x, 1) L ∞ (Ω)  ε locally uniformly as sn → +∞. This implies that there is N > 0 such that v n (x, 1)− w ˜ ) gives us that for all n  N. Maximum principle for e −μ(t −1) ( v n − w

   v n (x, t ) − w ˜ (x, t )  εe μ(t −1) ,

∀(x, t ) ∈ Ω × [1, +∞)

˜ ) satisfies since e −μ(t −1) ( v n − w

    M− D 2 z − zt  0  M+ D 2 z − zt

in Ω × (1, +∞)

˜ ) = 0 on ∂Ω × [1, +∞). We apply the regularity theory to e −μ(t −1) ( v n − w ˜ ) to and e −μ(t −1) ( v n − w estimate

   ∇x v n (x, 2) − w ˜ (x, 2) L ∞ (Ω)  C o e μ ε , ˜ (x, 2) = 0 for where the uniform constant C o > 0 depends only on λ, Λ, n and Ω . Since v n (x, 2) − w ˜ (x, 2) L ∞ (Ω)  e μ ε , and ∇x ( v n (x, 2) − w ˜ (x, 2)) L ∞ (Ω)  C o e μ ε , we deduce that x ∈ ∂Ω , v n (x, 2) − w

   v n (x, 2) − w ˜ (x, 2)  C˜ εϕ (x) in Ω for some uniform constant C˜ > 0 depending only on C o , Ω , and

˜ (x, 2) + C˜ εϕ (x)  v n (x, 2) = v (x, 2 + sn )  w



ϕ . Therefore we have 

γ ∗ + ε + C˜ ε ϕ (x) in Ω,

for some large sn > 0. By using maximum principle for u (x, t ) − e −μt (γ ∗ + ε + C˜ ε )ϕ (x), we conclude that

v (x, t ) = e μt u (x, t ) 





γ ∗ + ε + C˜ ε ϕ (x) for t  2 + sn ,

and it follows that

w





γ ∗ + ε + C˜ ε ϕ for all w ∈ A.

Since ε is arbitrary and C˜ is uniform, we have that w  γ ∗ ϕ for all w ∈ A. ˜ ≡ γ ∗ ϕ for some w ˜ ∈ A. By the Second, we show that A has only one element. Assume that w ˜ Now we set definition of A, we can find a sequence of functions { v (·, · + sn )} whose limit is w. ˜ (x, t ) and u 2 (x, t ) := γ ∗ ϕ (x)e −μt . Then u 1 − u 2 satisfies u 1 (x, t ) := e −μt w



    M− D 2 z − zt  0  M+ D 2 z − zt

in Ω × (0, +∞),

z=0

on ∂Ω × (0, +∞).

˜ (·, 0)  γ ∗ ϕ . Indeed, if not, the uniqueness says that w ˜ ≡ γ ∗ ϕ in Ω × It is easy to check that w (0, +∞), which is a contradiction. Thus the strong maximum principle and Hopf ’s Lemma imply that u 2 (x, 1) − u 1 (x, 1) > 0 in Ω and u 2 (x, 1) − u 1 (x, 1)  δ ϕ (x)

in Ω

˜ (x, 1)  (γ ∗ − δ e μ )ϕ (x) in Ω . Therefore we have that for small δ > 0, namely, w

˜ (x, t + 1)  e μ(t +1) u 1 (x, t + 1) = w





γ ∗ − δ eμ ϕ (x) in Ω × (0, +∞)

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from the comparison principle. Setting tn := sn + 1, we get

˜ (x, t + 1) v (x, t + tn ) → w

locally uniformly in Ω × [0, +∞),

˜ (x, t + 1)  (γ ∗ − δ e μ )ϕ (x). as n → +∞, which is a contradiction to the definition of γ ∗ since w Therefore we conclude that A = {γ ∗ ϕ }. Now we take sn = n for n ∈ N. Then for ε > 0, we can choose N ∈ N such that if n  N , then v (·, · + n) − γ ∗ ϕ C 0 (Ω×[0,1])  ε , which means

   v (x, t ) − γ ∗ ϕ (x) This finishes the proof.

L ∞ (Ω×[ N ,+∞))

 ε.

2

Under the additional assumption that F is concave, we obtain the following corollary. Corollary 4.6. Suppose that F satisfies ( F 1), ( F 2) and ( F 3). Let u, v and ϕ be the functions in Proposition 4.5. Then we have

   v (x, t ) − γ ∗ ϕ (x)

C xk (Ω)

→ 0 as t → +∞

for k = 1, 2, where γ ∗ > 0 is the constant in Proposition 4.5. Proof. Since F is concave, the eigenfunction ϕ is of C 2,α (Ω) and u and v also belong to C 2,α (Ω × (0, +∞)). Moreover, we have the uniform C 2,α -estimate:





sup v (·, · + s)C 2,α (Ω×[0,+∞)) < +∞ s1

(4.10)

since v is bounded from Corollary 4.3. Arzelà–Ascoli Theorem says that for any {sn } such that sn → +∞, there is a subsequence {snk } satisfying

⎧ ∗ ⎪ ⎨ v nk := v (·, · + snk ) → γ ϕ , ∇x v nk → γ ∗ ∇ ϕ , ⎪ ⎩ 2 D x v nk → γ ∗ D 2 ϕ locally uniformly in Ω × [0, +∞) as nk → +∞, since v (·, t ) converges uniformly to the unique limit γ ∗ ϕ from Proposition 4.5. Therefore, as t → +∞, ∇x v (·, t ) and D 2x v (·, t ) converge to γ ∗ ∇φ and γ ∗ D 2 φ in Ω, respectively, and then the uniform convergence follows from the uniform C 2,α -estimate (4.10). 2 4.2. Log-concavity In this subsection, we study log-concavity of solutions of (4.7) and (EV) provided that a smooth bounded domain Ω is convex and the operator F satisfies ( F 1), ( F 2) and ( F 3). First, let us approximate the operator with smooth operators as follows. Lemma 4.7. Suppose that F satisfies ( F 1), ( F 2) and ( F 3). Then there is a smooth operator F ε : Sn×n → R, which converges to F uniformly and satisfies ( F 1), ( F 3) and

 ε  √  F ( M ) M i j − F ε ( M )  nΛε for M = ( M i j ) ∈ Sn×n , ij ε where F iεj ( M ) := ∂∂ pF ( M ). ij

(4.11)

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Proof. We extend F to Rn by F ( M ) = F ( M +2M ). Then we can show that F is Lipschitz continuous t

2

2

in Rn with a Lipschitz constant

M+ ( N ) = Λ tr( N ) 

√

√

nΛ by using the uniform ellipticity of F , ( F 1), and the fact that

n

1

2 2 n×n . i , j =1 N i j ) for 0  N = ( N i j ) ∈ S

nΛ(

Let ψ ∈ C o∞ (Rn ) be a standard mollifier with 2

ψε ( Z ) =

1

ε

n2

ψ( εZ ). Define F ε as



Rn

2

ψ( Z ) d Z = 1 and supp(ψ) ⊂ B 1 (0) and let

 F ε ( Z ) := F ∗ ψε ( Z ) =

F ( Z − Y )ψε (Y ) dY .

2 Rn

Then it is easy to show that F ε is smooth, uniformly elliptic (with the same ellipticity constants λ, Λ) and concave. We can also show that F ε satisfies F ε ( M ) = F ε ( M t ) and

 ε  √  F ( M ) − F ( M )  nΛε 2

since F is Lipschitz continuous in Rn with a Lipschitz constant to F . Now, it remains to show that for any M = ( M i j ) ∈ Sn×n ,

√

nΛ. Thus F ε converges uniformly

 ε    √  F ( M ) M i j − F ε ( M ) =  D F ε ( M ) · M − F ε ( M )  nΛε . ij

Since F is Lipschitz continuous, F is differentiable almost everywhere from Rademacher’s Theorem. √ F ((1+t ) Z )− F ( Z ) = F ( Z ) for Moreover, we have D F L ∞ (Rn2 )  nΛ. The condition ( F 2) implies that t 2

Z ∈ Rn and t > 0 and hence

D F (Z) · Z = F (Z)

2

a.e. Z ∈ Rn .

Thus we have

 D F ε(Z ) · Z − F ε(Z ) = Rn





D F (Y ) · Z − F (Y ) ψε ( Z − Y ) dY

2



=

D F (Y ) · ( Z − Y )ψε ( Z − Y ) dY n2

R

and then | D F ε ( Z )· Z − F ε ( Z )| 

√

nΛε for M ∈ Sn×n .

√

2

nΛε for Z ∈ Rn . Therefore we conclude that | F iεj ( M ) M i j − F ε ( M )| 

2

Remark 4.8. (i) If a differentiable operator F satisfies ( F 2), F should be linear. If F also satisfies ( F 1), then F becomes Laplacian  after a suitable transformation. (ii) Let σk ( D 2 u ) := i 1 <···
F ( D 2 u ) := σk ( D 2 u ) k satisfies the conditions ( F 2) and ( F 3). Now, we prove that log-concavity is preserved under the parabolic flow of (4.7) when initial data has such property.

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Lemma 4.9. Suppose that F satisfies ( F 1), ( F 2) and ( F 3) and that Ω is strictly convex. Let u be the solution of (4.7) with initial data u o ∈ Cb (Ω). If log u o is concave, then the solution u (x, t ) is log-concave in the x variables for all t ∈ (0, +∞), i.e.,

D 2x log u  0

in Ω × (0, +∞).

Proof. (i) First, we prove the preservation of log-concavity assuming that u o ∈ C 2,γ (Ω) ∩ Cb (Ω) (0 < γ < 1), D 2x log uo  0 in Ω , and F ( D 2 uo ) = 0 on ∂Ω . We approximate F by F ε as Lemma 4.7 and we may assume that F ε (0) = 0 by subtracting F ε (0) to F ε . We also approximate u o by u oε ∈ C 2,γ (Ω) ∩ Cb (Ω) for small ε > 0 such that

u oε → u o

in C 2,γ (Ω),

D 2x log u oε  ε I

in Ω,

and





F ε D 2 u oε = 0 on ∂Ω.

In fact, let u oε be the solution of









F ε D 2 u oε = F D 2 u o



u oε = 0

in Ω, on ∂Ω.

Then we have the uniform global C 2,γ -estimate for u oε (0 < γ < 1) from [6] so we construct such initial data u oε from Arzelà–Ascoli Theorem. Indeed, we first note that u oε  δo u o uniformly in Ω for some δo > 0 since u o ∈ Cb (Ω). Then the argument below, (4.14) says that there is a uniform η > 0 such that D 2 log u oε  0 on Ω \ Ω(−η) and then uniform C 2 convergence in Ω(−η) , up to a subsequence, implies that D 2 log u oε  ε I on Ω \ Ω(−η) for small ε > 0. Thus we deduce that D 2x log u oε  ε I in Ω for small ε > 0. Let u ε be the solution of (4.7) with the operator F ε and initial data u oε . Then we have the uniform global C 2,γ -estimate for u ε (0 < γ < 1) from Theorem 3.2 in [26]; for a fixed T > 0,

 ε u 

C 2,γ (Ω×[0, T ])


uniformly.

(4.12)

The uniform C 2,γ -estimate gives the uniform convergence of u ε to u in C 2 (Ω × [0, T ]) by Arzelà– Ascoli Theorem since the family of viscosity solutions is closed in the topology of local uniform convergence, where we recall that u is the solution of (4.7) with the operator F and initial data u o . Since F ε  M− , we have the uniform lower bound from the comparison principle

u ε  u˜ > 0

in Ω × (0, +∞),

where u˜ is the solution of (4.7) with Pucci’s operator M− and initial data δo u o . We also observe that |∇x u ε (x, t )| > co > 0 uniformly for (x, t ) ∈ ∂Ω × [0, T ] for small co > 0 using Lemma 4.1 with Pucci’s operator M− . To show log-concavity of u, we consider an approximating solution u ε and we set

g ε := log u ε , which is finite and smooth in Ω and takes the value g ε = −∞ on ∂Ω × [0, +∞). The function g ε satisfies the equation

   ∂t g = e − g F ε e g D 2 g + D g D g t in Ω × (0, +∞). To estimate the maximum of its second derivatives, we define, for a given 0 < δ < 1,

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ε ( y , t ) + ψ(t ), Z (t ) := sup sup g ,ββ y ∈Ω |e β |=1

√

where e β is a unit vector in Rn and ψ(t ) := −δ − δ tan(3K δt ). The constant K > 0, independent of ε > 0 and δ > 0, will be chosen later. Now, fix T > 0 and suppose that there exists t o ∈ [0, min( π√ , T )] ⊂ [0, T ] such that Z (t o ) = 0. 12K

δ

We may assume that t o is the first time for Z to vanish and

Z (to ) = g ,εαα (xo , to ) + ψ(to ) = 0 for some direction e α ∈ Rn with |e α | = 1 and some point xo ∈ Ω . Then the unit vector e α is an eigen-direction of the Hessian matrix D 2x g ε (xo , to ), which implies that

g ,εα β (xo , to ) = 0 for β = α , using orthonormal coordinates in which e α is taken as one of the coordinate axes. We notice that Z (0)  ε − δ < 0 if 0 < ε < δ from the assumption on the initial data u o , which implies that to > 0. We claim that xo is an interior point of Ω by proving that for any x˜ ∈ ∂Ω , t > 0

g ,εαα (x, t ) =

u ε u ε,αα − (u ε,α )2

(u ε )2

→ −∞ as Ω  x → x˜ ∈ ∂Ω.

(4.13)

The above inequality holds when e α is not a tangential direction to ∂Ω at x˜ , since | D 2 u ε | is uniformly bounded and u ε = 0 on ∂Ω , |∇ u ε | > 0 on ∂Ω by Hopf ’s Lemma. If e α is a tangential direction to ∂Ω at x˜ , we take a coordinate system such that x˜ = 0 and the tangent plane to ∂Ω at x˜ = 0 ∈ ∂Ω is xn = 0 with en being the inner normal vector. Let the boundary be given locally by the equation xn = f (x ) for x = (x1 , . . . , xn−1 ) ∈ Rn−1 . We introduce the change of variables:

y i = xi

(i = 1, . . . , n − 1),

 

yn = xn − f x ,

v ( y , t ) = u ε (x, t ).

Then along any tangent direction e τ to ∂Ω at x˜ , we have

 

 

 

u ε,τ τ (x, t ) = v ,τ τ ( y , t ) − 2v ,nτ ( y , t ) f τ x + v ,nn ( y , t ) f τ2 x − v ,n ( y , t ) f ,τ τ x . Using the fact that v ,ii (0, t ) = 0 from the zero boundary condition and f ,i (0) = 0 for i = 1, . . . , n − 1, we obtain

u ε,τ τ (˜x, t ) = u ε,τ τ (0, t ) = − v ,n (0, t ) f ,τ τ (0) = −u ε,n (˜x, t ) f ,τ τ (0) < 0 from Hopf ’s Lemma since f ,τ τ (0) > 0. We note that f ,τ τ (0) > 0 for a tangent vector e τ since Ω is strictly convex and that u ετ (˜x, t ) = 0 from the zero boundary condition. Thus g ,ετ τ (x, t ) tends to −∞ when x ∈ Ω goes to x˜ ∈ ∂Ω for any tangential vector e τ to ∂Ω at x˜ ∈ ∂Ω , so this is true for the tangential vector e α . Therefore we have proved (4.13). Moreover, from the uniform C 2,γ -estimate of u ε , (4.12), we can find a small η > 0, independent of ε, δ > 0, such that

g ,εαα (x, t )  0 for (x, t ) ∈ (Ω \ Ω(−η) ) × [0, T ], where Ω(−η) = {x ∈ Ω : dist(x, ∂Ω) > η}. Indeed, we first observe that for any (x, t ) ∈ ∂Ω × [0, T ],

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

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u ε,τ τ (x, t ) < −co κo < 0 for any tangential direction τ to ∂Ω at x, with κo > 0, the lower bound of the curvature of ∂Ω , from the above argument, since |∇x u ε | > co > 0 uniformly on ∂Ω × [0, T ]. If η > 0 is small enough, then for x ∈ Ω \ Ω(−η) , there exists a unique x˜ ∈ ∂Ω such that |x − x˜ | = dist(x, ∂Ω). For each x˜ ∈ ∂Ω , let ν (˜x) be the outer normal vector to ∂Ω at x˜ ∈ ∂Ω and let τ (˜x) be the unit vector such that τ (˜x) ⊥ ν (˜x) and

e α := β1 (˜x)τ (˜x) + β2 (˜x)ν (˜x) with β12 + β22 = 1. We define ν (x) := ν (˜x) and τ (x) := τ (˜x) for each x ∈ Ω \ Ω(−η) , where x˜ is the unique boundary point of Ω such that |x − x˜ | = dist(x, ∂Ω). Then the uniform C 2,γ -estimate implies that there is a uniform small η > 0 such that

 ε  u (x, t ) > co , ν 2

u ε,τ τ (x, t ) < −

co κo 2

on (Ω \ Ω(−η) ) × [0, T ].

For each x ∈ Ω \ Ω(−η) and t ∈ [0, T ] we have





u ε,τ (x, t ) = u ε,τ (˜x) (x, t ) = u ε,τ (˜x) (˜x, t ) + ∇ u ε,τ (˜x) x∗ , t · (x − x˜ )

 0 + C o |x − x˜ | = C o dist(x, ∂Ω)  C˜ o u ε (x, t ) for some x∗ ∈ Ω and for uniform constants 0 < C o < C˜ o from the uniform C 2,γ -estimates since |∇x u ε | > co > 0 uniformly on ∂Ω × [0, T ]. Thus for x ∈ Ω \ Ω(−η) , we have

 ε 2 ε  2 u g ,αα (x, t ) = u ε u ε,αα (x, t ) − u ε,α (x, t )  = u ε β12 (˜x)u ε,τ τ (x, t ) + 2β1 (˜x)β2 (˜x)u ε,τ ν (x, t ) + β22 (˜x)u ε,νν (x, t )  2  2 − β22 (˜x) u εν (x, t ) − β12 (˜x) u ετ (x, t ) − 2β1 (˜x)β2 (˜x)u ετ (x, t )u εν (x, t )   u ε β12 (˜x)u ε,τ τ (x, t ) + 2β1 (˜x)β2 (˜x)u ε,τ ν (x, t ) + β22 (˜x)u ε,νν (x, t ) −

β22 2

 2  2 (˜x) u εν (x, t ) + β12 (˜x) u ετ (x, t )



 2 co κo 2 co κo 2 c2  u ε (x, t ) − β1 + β1 + C o β22 − o β22 + C˜ o2 u ε (x, t ) β12 2

4

8

   2  co κo co 2 ε ε 2 ε ˜ = −C o u (x, t ) − u (x, t ) β1 − − C o u (x, t ) β22 , 8 4C˜ o2

(4.14)

for uniform C˜ o > C o > 0, where we use Young’s inequality and the uniform C 2,γ -estimate. By using the uniform C 2,γ -estimate again, we choose η > 0 sufficiently small to deduce that g ,εαα  0

in (Ω \ Ω(−η) ) × [0, T ]. Therefore, we conclude that the maximum point xo belongs to Ω (−η) since g ,εαα (xo , to ) = −ψ(to ) > δ > 0. Next, we look at the evolution equation of g ,εαα , which is given by the equation as below:

∂t g ,αα = F iεj · ( D i j g ,αα + D i g ,αα D j g + D i g D j g ,αα + 2D i g ,α D j g ,α )     + g ,2α − g ,αα e − g F ε e g ( D i j g + D i g D j g ) − F iεj · ( D i j g + D i g D j g )   + e − g F iεj ,kl · e g ( D i j g + D i g D j g ) α e g ( D kl g + D k g D l g ) α ,

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S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306 ε

2

ε

ε

ε

where F iεj = ∂∂ pF (e g ( D 2 g ε + D g ε ( D g ε )t )) and F iεj ,kl = ∂ p∂ ∂F p (e g ( D 2 g ε + D g ε ( D g ε )t )). Since F ε satij ij kl √ √ isfies ( F 3) and (4.11) with the constant 2 nΛε instead of nΛε , it follows that ε  F ε ·  D g ε + D g ε D g ε + D g ε D g ε + 2D g ε D g ε  ∂t g αα i j ,αα i ,αα j i j ,αα i ,α j ,α ij   √   ε 2 + 2 nΛe − g  g ,εα − g ,εαα ε .

At the point of maximum (xo , to ), we see that

g ,εαα = −ψ > δ > 0,

∇x g ,εαα = 0,

D 2x g ,εαα  0 and

g ,εα β = 0,

∀β = α .

Thus using the ellipticity condition of F ε , we have at the point of maximum (xo , to ), ε  g ε 2 + 2√nΛe − g ε  g ε 2 + g ε ε ∂t g ,εαα  2F αα ,αα ,α ,αα √ |u ε,αα |  ε 2  2Λ g ,αα + 2 nΛ ε 2 ε (u )  ε 2  2Λ g ,αα + K ε ,

√

for a uniform constant K > 0, where we choose a uniform K > 0 bigger than 2 nΛ(1 + | D 2 uε | supΩ (−η) ×[0, T ] (uxε )2 ) from the uniform C 2,γ -estimates for u ε .

On the other hand, when the supremum of Z (t ) − ψ(t ) = sup y ∈Ω sup|eβ |=1 g ,ββ ( y , t ) is achieved at a point x(t ) ∈ Ω with a unit vector e β(t ) at each time t > 0, we can check that ∇x g ,β(t )β(t ) = 0 and g ,β(t )β  (t ) = 0 at the point (x(t ), t ) using |e β (t )|2 = 1. Thus, we have that at the first time t o > 0,

0  Z  (to ) = ∂t g ,εαα (xo , to ) + ψt (to )

   ψt + 2Λψ 2 + K ε  ψt + K ψ 2 + ε .

(4.15)

But, it is easy to check that if 0 < ε < δ 2 ,

√   3K (−δ 3/2 + δ 2 ) π < 0 for 3K δt < , ψt + K ψ 2 + ε < √ 2 cos(3K δt ) which is a contradiction to (4.15). Therefore, the function Z never reaches 0 for t ∈ [0, min(

π√ , T )]

12K

δ

and hence we obtain that



 sup sup ∂αα log u ε ( y , t ) < −ψ(t )  2δ

y ∈Ω |e α |=1

for t ∈ 0, min

for 0 < ε < δ 2 . Using the uniform C 2,γ -estimates of u ε , we let obtain

π



√ ,T 12K δ

ε go to 0 and then let δ go to 0 to

D 2x log u  0 in Ω × [0, T ]. Therefore, u (x, t ) is log-concave in the x variables in Ω × (0, +∞) since T is arbitrary. (ii) For general initial data u o ∈ Cb (Ω), we will show that there is a sequence of log-concave functions u oj ∈ C 2,γ (Ω) ∩ Cb (Ω) satisfying that F ( D 2 u oj ) = 0 on ∂Ω , which converge to u o uniformly in Ω .

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

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First, we may assume that u o ∈ C ∞ (Ω) ∩ Cb (Ω) is strictly log-concave. Indeed, we perform a mollification to obtain an approximating sequence u oj ∈ C ∞ (Ω) of log-concave functions, which converges to u o uniformly in Ω . We modify u oj to make it strictly log-concave





u˜ oj (x) := u oj (x) exp −c j |x|2 , where c j > 0 converges to 0 as j → +∞. For a strictly log-concave u o ∈ C ∞ (Ω) ∩ Cb (Ω), we consider

 





F D 2 u oj = ξ j · F D 2 u o

u oj = 0



in Ω, on ∂Ω,

where 0  ξ j  1 satisfies that ξ j ∈ C o∞ (Ω), and ξ j ≡ 1 in Ω(−1/ j ) for Ω(−δ) := {x ∈ Ω : dist(x, ∂Ω) > δ}. Since we have the global uniform C 1,α -estimate (0 < α < 1) from [6], u oj converges to u o in C 1,α (Ω) as j → +∞, up to a subsequence. Since u oj ∈ C 2,γ (Ω), and F ( D 2 u oj ) = 0 on ∂Ω , it remains to show that u oj is log-concave for large j in order to obtain the desired initial data u oj converging to u o uniformly. We use the uniform global C 1,α -estimate and the scaling property with a similar argument as in the proof of Theorem 3.4 to show that there is a uniform constant ηo > 0 such that

 2   D u oj (x)  C dist(x, ∂Ω)−1+α for x ∈ Ω \ Ω(−η ) , o where

ηo and C > 0 are uniform with respect to j. By choosing a uniform small 0 < η < ηo , we have

u oj (x) D 2 u oj (x) − ∇ u oj (x)∇ utoj (x)  C dist(x, ∂Ω)1+(−1+α ) I − δo2 I  0 for x ∈ Ω \ Ω(−η) for uniform C > 0 and δo > 0 since u o , u oj ∈ Cb (Ω). For Ω(−η) , we have the uniform interior C 2,γ estimate of u oj if j is large enough. So we have that D 2 log u oj  0 in Ω(−η) since log u o is strictly concave. Therefore, we deduce that u oj is log-concave. Thus we have proved that for a given logconcave function u o ∈ Cb (Ω), there is a sequence of log-concave functions u oj ∈ C 2,γ (Ω) ∩ Cb (Ω) satisfying that F ( D 2 u oj ) = 0 on ∂Ω , which converge to u o uniformly in Ω . Let u oj converge to u o uniformly in Ω and let u j and u be the solution of (4.7) with the operator F and initial data u oj and u o , respectively. From the maximum principle for u j − u, we have

u j − u L ∞ (Ω×[0,+∞))  u oj − u o L ∞ (Ω) → 0 as j → +∞ since u j − u satisfies M− ( D 2 v ) − v t  0  M+ ( D 2 v ) − v t in Ω × (0, +∞). Then log-concavity of u j from (i)

1 2



log u j (x, t ) + log u j ( y , t ) − log u j



x+ y 2

 , t  0 for x, y ∈ Ω, t ∈ (0, +∞),

is preserved under the uniform convergence. Therefore, we conclude that u (x, t ) is log-concave in the x variables for any t > 0. 2 Corollary 4.10. Suppose that F satisfies ( F 1), ( F 2) and ( F 3) and that Ω is convex. Let u be the solution of (4.7) with initial data u o ∈ Cb (Ω). If u o is log-concave, then the solution u (x, t ) is log-concave in the x variables for t > 0.

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Proof. We approximate Ω by Ω j which are strictly convex, smooth and bounded domains in Rn and approximate initial data u o by u oj ∈ Cb (Ω j ). Let u j be the solution of (4.7) with the operator F and initial data u oj in Ω j × (0, +∞). Then we have uniform local Hölder estimates for u j in Ω × (0, +∞), namely, u j C α ( K ×[t0 ,t1 ]) < C uniformly for each compact subset K × [t 0 , t 1 ] of Ω × (0, +∞). We can also check that u j converges to u pointwise in Ω × [0, +∞) by using the maximum principle. So we have uniform convergence of u j to u locally in Ω × (0, +∞), up to a subsequence, from Arzelà–Ascoli Theorem. Therefore for any K × [t 0 , t 1 ] ⊂ Ω × (0, +∞), we let j go to +∞ to obtain

1 2





log u (x, t ) + log u ( y , t ) − log u

which completes the proof.

x+ y 2

 , t  0 for x, y ∈ K , t ∈ [t 0 , t 1 ],

2

Remark 4.11. (i) We note that any concave function in a convex domain Ω is log-concave. (ii) It is well-known that the distance function dist(x, ∂Ω) is concave for a convex domain Ω , so Lemma 4.9 and Corollary 4.10 are not void. Corollary 4.12 (Log-concavity). Suppose that F satisfies ( F 1), ( F 2) and ( F 3) and that Ω is convex. Then, the positive eigenfunction ϕ (x) in Theorem 3.1 is log-concave, i.e., D 2 log ϕ (x)  0 for x ∈ Ω . Proof. We take the distance function as an initial data of the uniformly parabolic equation (4.7). Then Corollary 4.10 yields that for x, y ∈ Ω and t > 0,

1 2





log u (x, t ) + log u ( y , t ) − log u

From the uniform convergence of e μt u (·, t ) to

x+ y 2

 , t  0.

γ ∗ ϕ in Proposition 4.5, we conclude that

 1 x+ y log ϕ (x) + log ϕ ( y ) − log ϕ 2 2

  0 for x, y ∈ Ω.

2

For a differentiable operator, the foregoing is a classical result [23] for a smooth bounded domain which is strictly convex (see Remark 4.8 (i)). Lemma 4.13 (Strict log-concavity). Suppose that F satisfies ( F 1), ( F 2) and is differentiable and that Ω is strictly convex. Then the positive eigenfunction ϕ of (EV) is strictly log-concave, i.e., there exists a constant c 1 > 0 such that

D 2 log ϕ  −c 1 I. Theorem 4.14 (Eventual log-concavity). Under the same hypothesis on the operator and the domain as Lemma 4.13, let u be the solution of (4.7) with a nonzero nonnegative initial data u o ∈ C 0 (Ω). Then, u (x, t ) is strictly log-concave in the x variables for large t > 0, i.e., for any ε > 0, there is t o = to (u o , ε ) > 0 such that

D 2x log u (x, t )  −(c 1 − ε )I where c 1 > 0 is the constant of Lemma 4.13.

for x ∈ Ω, t  to ,

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

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5. Degenerate parabolic fully nonlinear equation In this section, we study the fully nonlinear degenerate parabolic equation:

⎧   2 m ⎪ ⎨ F D u − ∂t u = 0 in Ω × (0, +∞), u (·, 0) = u o in Ω, ⎪ ⎩ u=0 on ∂Ω × (0, +∞),

(5.16)

in the range of exponents m > 1. We always assume that Ω is a smooth bounded domain of Rn and F satisfies ( F 1) throughout this section. We also assume that um o belongs to

  Cb (Ω) := h ∈ C o (Ω)  co dist(x, ∂Ω)  h(x)  C o dist(x, ∂Ω) for some 0 < co  C o < +∞ , which means that um o ∈ Cb (Ω) has nontrivial bounded gradient on ∂Ω . By setting w := um , we transform the problem (5.16) to

⎧  2  1− 1 ⎪ ⎨ mw m F D w − ∂t w = 0 in Ω × (0, +∞), m ⎪ w (·, 0) = w o = u o ∈ Cb (Ω) in Ω, ⎩ w =0 on ∂Ω × (0, +∞).

(5.17)

We also introduce the pressure in the form

v=

m m−1

u m −1 ,

which is defined as a physical quantity in a model of flow of gases in porous media. If F satisfies ( F 2), the pressure v solves

⎧   v t = F (m − 1) v D 2 v + D v D v t in Ω × (0, +∞), ⎪ ⎪ ⎨ m v (·, 0) = v o = in Ω, u m −1 ⎪ m−1 o ⎪ ⎩ v =0 on ∂Ω × (0, +∞).

(5.18)

Before analyzing asymptotic behaviors of degenerate parabolic flows, let us state the regularity of the solution to (5.16). Proposition 5.1 (Regularity for m > 1). Suppose that F satisfies ( F 1) and let u be the solution of (5.16). Then the following hold. (i) If u o ∈ C 0 (Ω) is nonzero and nonnegative, then there is a time t o > 0 such that

u (x, t ) > 0

in Ω × [to , +∞).

(ii) If um o ∈ Cb (Ω), then we have for (x, t ) ∈ Ω × (0, +∞), 1

1

co (τ1 + t )− m−1 dist(x, ∂Ω) m  u (x, t )  C o (τ2 + t )− m−1 dist(x, ∂Ω) m 1

1

(5.19)

for some constants τ1 , τ2 > 0 depending on u o and for uniform constants co , C o > 0. Moreover, let Ω   Ω be a subset of Ω and let K := Ω  × [s, T ]  Ω × (0, +∞) for 0 < s < T . (a) um is of C 1,β ( K ) for some 0 < β < 1. (b) If F is concave, um is of C 1,1 ( K ).

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(c) If um ∈ C 1,1 ( K ) and F is concave, then um is of C 2,β ( K ) for some 0 < β < 1. (d) If um ∈ C 2,β ( K ) and F ∈ C ∞ , then u is of C ∞ ( K ). Proof. (i) We construct a sub-solution of F ( D 2 um ) − ut = 0, whose support expands in time. For c > 0, define





|x|2 V (x, t ) := t −α c − k β , t + where that

α :=

n(m−1)Λ , 2λ+n(m−1)Λ

β :=

2λ , 2λ+n(m−1)Λ

k =: 2(2λ+n(1m−1)Λ) and f + := max{ f , 0}. It is easy to check

  M− (m − 1) V D 2 V + D V D V t − V t      M− (m − 1) V D 2 V + M− D V D V t − V t = 0 in { V > 0} and the support of V expands in time. Thus V is a sub-solution of M− ((m − 1) v D 2 v + D v D v t ) − v t = 0 as long as supp( V (·, t )) ⊂ Ω . We define

U (x, t ) :=

m−1 m

 1

 1  m−1 1

m − 1 m −1 − α |x|2 m−1 m − 1 = t c −k β V (x, t ) m

t

+

and then U is a sub-solution of M− ( D 2 um ) − ut = 0 as long as supp(U (·, t )) ⊂ Ω . Since F  M− , U is also a sub-solution of F ( D 2 um ) − ut = 0 as long as supp(U (·, t )) ⊂ Ω . We note that the support 1

of U is compact and expands in time. We also recall a separable solution (φ − ) m (τ + t )− m−1 (for τ > 0) of (5.16) with Pucci’s operator M− in place of F , where φ − is the positive solution of (NLEV) in Theorem 3.4 with Pucci’s operator M− . So, by using a similar argument in Lemma 4.2, we obtain that u is positive for a large time t  t o > 0. (ii) Let φ + and φ − be the positive solution of (NLEV) in Theorem 3.4 with Pucci’s operator M+ and M− , respectively. Since φ ± ∈ C 0,1 (Ω) and infx∈∂Ω |∇φ ± (x)| > δ > 0 for some δ > 0, there exist C o+ , co+ , C o− , co− > 0 such that 1

co± dist(x, ∂Ω)  φ ± (x)  C o± dist(x, ∂Ω) We can choose

for x ∈ Ω.

τ1 > 0 and τ2 > 0 such that for any x ∈ Ω , − mm −1

φ − (x)τ1

− mm −1

 τ1

C o− dist(x, ∂Ω)  um o (x)

− mm −1 +

− mm −1

co dist(x, ∂Ω)  φ + (x)τ2

 τ2

1

− m−1 ± m since um (i = 1, 2) is a separable solution of (5.16) with Pucci’s o ∈ Cb (Ω). Since (φ ) (x)(τi + t ) ± − operator M , respectively, and M  F  M+ , the comparison principle implies that for (x, t ) ∈ Ω × (0, +∞), 1



φ−

 m1

 1 1 1 (x)(τ1 + t )− m−1  u (x, t )  φ + m (x)(τ2 + t )− m−1 .

Thus we have for (x, t ) ∈ Ω × (0, +∞),



co−

 m1

 1 1 1 1 1 (τ1 + t )− m−1 dist(x, ∂Ω) m  u (x, t )  C o+ m (τ2 + t )− m−1 dist(x, ∂Ω) m ,

which proves (5.19).

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

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Let w := um . For a compact subset K := Ω  ×[s, T ]  Ω ×(0, +∞), we can choose a subdomain Ω  of Ω such that Ω   Ω   Ω . Let K˜ := Ω  × [s/2, T ]. From (5.19), there exist 0 < c  C < +∞ depending on K˜ such that

in K˜ ,

0 < c  w = um  C < +∞

which implies that the operator mw 1− m (x) F (·) in (5.17) is uniformly elliptic in K˜ . So the interior estimates in K , (a)–(d), follow from the standard regularity theory of uniformly parabolic equations [25, 26,19] (see also Theorem 4.4). 2 1

5.1. Asymptotic behavior In this subsection, we study the asymptotic behavior of the solution of (5.16) in the range of exponents m > 1 when t → +∞. Proposition 5.2. Suppose that F satisfies ( F 1) and ( F 2). Let u be the solution of (5.16) with um o ∈ Cb (Ω) and let φ be the solution of (NLEV) in Theorem 3.4. Set W (x, t ) :=





t m−1 um (x, t ) − W (x, t ) → 0 m

(i)

φ(x)

m

(1+t ) m−1

. Then, we have

uniformly for x ∈ Ω, as t → +∞,

(ii) for (x, t ) ∈ Ω × [1, +∞), m

co dist(x, ∂Ω) < t m−1 um (x, t ) < C o dist(x, ∂Ω), (iii) for (x, t ) ∈ Ω × [2, +∞),





t m−1 ∇ um (x, t ) < C o , m

where the uniform constants 0 < co < C o depend on u o . Proof. (i) We recall that φ ∈ C 0,1 (Ω) and inf∂Ω |∇φ| > δ > 0 for some δ > 0. We can choose τ2 > 0 such that − mm −1

φ(x)τ1

− mm −1

 um o (x)  φ(x)τ2

τ1 >

for x ∈ Ω

as in the proof of (ii), Proposition 5.1 since um o ∈ Cb (Ω). The comparison principle implies that for (x, t ) ∈ Ω × (0, +∞), m

m

φ(x)(τ1 + t )− m−1  um (x, t )  φ(x)(τ2 + t )− m−1 m

since φ(x)(κ + t )− m−1 is a separable solution of (5.17) for any Ω × (0, +∞),







κ > 0. Thus, we have for (x, t ) ∈ 

t m−1 um (x, t ) − W (x, t )  t m−1 φ(x) · max (τi + t )− m−1 − (1 + t )− m−1 , i = 1, 2 m

m

   max φ(x) · max  x∈Ω m

m

m

m

m

t m −1 m

(τi + t ) m−1

−

t m −1 m

(1 + t ) m−1

and hence we deduce that limt →+∞ t m−1 um (·, t ) − W (·, t ) L ∞ (Ω) = 0.



   , i = 1, 2 

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(ii) Let w := um . In the proof of (i), we have for (x, t ) ∈ Ω × (0, +∞), m

m

φ(x)(τ1 + t )− m−1  w (x, t ) = um (x, t )  φ(x)(τ2 + t )− m−1 . Since φ ∈ C 0,1 (Ω) and inf∂Ω |∇φ| > δ > 0 for some δ > 0, we can find positive constants 0 < c 1 < C 1 < +∞ such that for (x, t ) ∈ Ω × (0, +∞),

c1

(1 + t )

m m −1

dist(x, ∂Ω)  w (x, t ) 

C1 m

(1 + t ) m−1

dist(x, ∂Ω)

and then for (x, t ) ∈ Ω × [1, +∞),

c1 2

m m −1

t

m m −1

dist(x, ∂Ω)  w (x, t ) = um (x, t ) 

C1

dist(x, ∂Ω).

m

(5.20)

t m −1

Therefore, (ii) follows. (iii) Let 0 < δo < 1 be a constant such that B δo (x) ⊂ Ω for x ∈ Ω(−δo ) := {x ∈ Ω : dist(x, ∂Ω) > δo }. Let (xo , to ) ∈ (Ω \ Ω(−δo ) ) × [2, +∞). For xo ∈ Ω \ Ω(−δo ) , we set dist(xo , ∂Ω) = 2σ . According to (5.20), it follows that for (x, t ) ∈ B σ (xo ) × [to /2, to ], m

c 2 σ  tom−1 w (x, t )  C 2 σ , m

(5.21)

m

where c 2 := c 1 2− m−1 and C 2 := 3C 1 2 m−1 . We define for (˜x, t˜) ∈ B 1 (0) × [σ −1−1/m − 1, σ −1−1/m ], m

˜ (˜x, t˜) := w where xo + σ x˜ ∈ B σ (xo ) and ˜ satisfies w 1

tom−1

σ





w xo + σ x˜ , σ 1+1/m to t˜ ,

σ 1+1/m to t˜ ∈ [to − σ 1+1/m to , to ] ⊂ [to /2, to ]. From the scaling property,





˜ t = 0 in B 1 (0) × ˜ 1− m F D 2 w ˜ −w mw





σ −1−1/m − 1, σ −1−1/m .

From (5.21), we have

˜  C2 c2  w

in B 1 (0) ×





σ −1−1/m − 1, σ −1−1/m ,

˜ solves a uniformly parabolic equation in B 1 (0) × [σ −1−1/m − 1, σ −1−1/m ] with which implies that w the ellipticity constants depending only on m, c 2 , C 2 , λ and Λ. From uniform gradient estimates for uniformly parabolic equations, Theorem 1.3 and Theorem 4.8 in [26], we obtain that

   ∇ w ˜ 0, σ −1−1/m  < C , where C > 0 depends only on m, c 2 , C 2 , λ and Λ. Therefore we deduce that m











˜ 0, σ −1−1/m  < C tom−1 ∇ w (xo , to ) = ∇ w and hence





t m−1 ∇ w (x, t ) < C m

uniformly for (x, t ) ∈ (Ω \ Ω(−δo ) ) × [2, +∞).

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

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For t o ∈ [2, +∞), (5.20) gives that m

c˜  tom−1 w (x, t )  C˜ m

for (x, t ) ∈ Ω(−δo /2) × [to /2, to ],

m

where c˜ := c 1 2− m−1 δo /2 and C˜ := C 1 2 m−1 diam(Ω). We define for (˜x, t˜) ∈ Ω(−δo /2) × [1/2, 1], m

˜ (˜x, t˜) := tom−1 w (˜x, to t˜), w ˜  C˜ in Ω(−δo /2) × [1/2, 1]. From the scaling property, w ˜ satisfies where to t˜ ∈ [to /2, to ] and c˜  w 1





˜ t = 0 in Ω(−δo /2) × [1/2, 1], ˜ 1− m F D 2 w ˜ −w mw which is uniformly parabolic in Ω(−δo /2) × [1/2, 1] with the ellipticity constants depending only on m, c˜ , C˜ , λ and Λ. Thus we have m









˜ (x, 1) < C tom−1 ∇ w (x, to ) = ∇ w

uniformly for x ∈ Ω(− 3 δ ) 4 0

from uniform gradient estimates, Theorem 1.3 and Theorem 4.8 in [26], where C > 0 depends only m on m, c˜ , C˜ , λ and Λ. Therefore, we conclude that t m−1 |∇ w (x, t )| is uniformly bounded in Ω(− 3 δ ) × 4 0

[2, +∞) and hence





t m−1 ∇ w (x, t ) < C m

uniformly in Ω × [2, +∞)

by putting together with the above argument.

2

Corollary 5.3. Under the same assumption of Proposition 5.2, we also assume that F is concave. For each compact subset K of Ω , we have

 m m  t m−1 u (·, t ) − φ 

C xk ( K )

→ 0 as t → +∞

for k = 0, 1, 2. Proof. Since

 m m      t m−1 u (x, t ) − φ(x)  t mm−1 um (x, t ) − W (x, t ) + t mm−1 W (x, t ) − φ(x)  

 mm−1    m  t − 1,  t m−1 um (x, t ) − W (x, t ) + φ(x) 1+t m

we have the uniform convergence of t m−1 um (·, t ) to the limit φ as t → +∞ from Proposition 5.2. For K  Ω , we can find a compact set K  such that K  K   Ω . Then there is a time T > 0 such that for (x, t ) ∈ K  × [ T , +∞),

1

m

− inf φ  t m−1 um (x, t ) − φ(x)  sup φ. 2

K

K

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Let w := um . Then we have

1 2

m

inf φ  t m−1 w (x, t )  2 sup φ. K

K

For t o ∈ [2T , +∞), we have m

m

c˜  tom−1 w (x, t ) = tom−1 um (x, t )  C˜ where c˜ :=

for (x, t ) ∈ K  × [to /2, to ],

m

inf K  φ and C˜ := 21+ m−1 sup K  φ . From (iii) in Proposition 5.2, we also have

1 2

m





tom−1 ∇ w (x, t ) < 2 m−1 C o m

uniformly for (x, t ) ∈ Ω × [to /2, to ],

and then for (x, t ) ∈ K  × [to /2, to ],

  mm−1 1− 1  m − 1  mm−1 − 1  mm−1  m−1 −1 m m ∇ t o w to w m to ∇ w (x, t )  (˜c ) m 2 m−1 C o , = m

m

where a uniform constant C o > 0 is in Proposition 5.2. By using a similar argument as in the proof of (iii), Proposition 5.2 and the concavity of F , we can apply Theorem 1.1 and Theorem 4.13 in [26], to deduce

 mm−1 m  to u (·, to )

C 2,α ( K )

 m  = tom−1 w (·, to )C 2,α ( K ) < C ,

where 0 < α < 1 and a uniform constant C > 0 depends only on m, c˜ , C˜ , C o , λ, Λ, K and K  . So we have proved

 m m  t m−1 u (·, t )

C 2,α ( K )

< C uniformly for t ∈ [2T , +∞). m

Now we use Arzelà–Ascoli Theorem and the uniform convergence of t m−1 um (·, t ) to the unique limit φ to conclude that

 m m  t m−1 u (·, t ) − φ 

C xk ( K )

→ 0 as t → +∞

for k = 1, 2. For details of the proof, we refer Corollary 4.6.

2

Corollary 5.4. Suppose that F satisfies ( F 1) and ( F 2). Let u be the solution of (5.16) with um o ∈ Cb (Ω). Set U (x, t ) :=

f (x)

1

(1+t ) m−1

, where f solves

⎧  2 m 1 ⎪ ⎪ ⎨ −F D f = m − 1 f f =0 ⎪ ⎪ ⎩ f >0

in Ω, on ∂Ω, in Ω.

Then, we have





t m−1 u (x, t ) − U (x, t ) → 0 1

uniformly for x ∈ Ω, as t → +∞,

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

3289

and hence





lim t m−1 u (·, t ) − f  L ∞ (Ω) = 0. 1

t →+∞

Proof. Let φ be the solution of (NLEV) in Theorem 3.4. Since φ = f m and (1 − r )m  1 − r m for 0  r  1, the result follows from Proposition 5.2. 2 5.2. Square-root concavity of the pressure m−1 be the pressure and let Let u be the solution of the problem (5.16) with um o ∈ Cb (Ω). Let v := u 2 v =: w . We prove the concavity of w in spatial variables for any t > 0 if the initial data u o has the

√

m−1

concavity of u o 2 , under some assumptions. The function w = v is a suitable quantity to perform geometrical investigation, which was demonstrated by Daskalopoulos, Hamilton and Lee in [10] for F = . First, let us approximate the problem (5.16) as follows: for 0 < η < 1,

⎧  2 m ⎪ in Ω × (0, ∞), ⎨ ∂t u η = F D u η uη = η on ∂Ω × (0, ∞), ⎪ ⎩ u η (·, 0) = u η,o  η in Ω.

(5.22)

Let g η := um η . Then g η satisfies the following problem:

⎧  1−1/m  2 ⎪ F D gη in Ω × (0, ∞), ⎨ ∂t g η = mg η g η = ηm on ∂Ω × (0, ∞), ⎪ ⎩ m m g η (·, 0) = g η,o = u η,o  η in Ω. We note that g η satisfies a uniformly parabolic equation in Ω × (0, +∞) for a fixed g η  ηm from the comparison principle. We assume that g η,o satisfies

1 2

go  g η,o − ηm  2go

(5.23)

η > 0 since

in Ω.

(5.24)

Lemma 5.5. Suppose that F satisfies ( F 1) and F (0) = 0. Let g η be the solution of (5.23) with the initial data g η,o ∈ C 0 (Ω) satisfying (5.24) for some go ∈ Cb (Ω). There are uniform positive constants c 0 , c 1 and K with respect to 0 < η < 1 and F such that

c 0 dist(x, ∂Ω)e − K t < g η (x, t ) − ηm < c 1 dist(x, ∂Ω),

∀(x, t ) ∈ Ω × [0, +∞),

and





0 < c 0 e − K t < ∇x g η (x, t ) < c 1 ,

∀(x, t ) ∈ ∂Ω × [0, +∞).

Proof. We establish a sub-solution and a super-solution of (5.23). From Theorem 3.1, let



  −M− D 2 ϕ − = μ− ϕ − in Ω,

ϕ− = 0

on ∂Ω,

ϕ − solve (EV)

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associated with the eigenvalue μ− > 0. We may assume that g η,o  ηm + ϕ − by multiplying a positive constant to ϕ − since g η,o − ηm  12 go and go ∈ Cb (Ω). We define







for K := μ− m 1 + ϕ −  L ∞ (Ω)

h(x, t ) := ηm + ϕ − (x)e − K t

1−1/m

> 0.

Then we have









mh1−1/m F D 2 h − ht  mh1−1/m M− D 2 h − ht

   = mh1−1/m e − K t M− D 2 ϕ − +

= mh

1−1/m − K t

e

ϕ

−



K ϕ− mh1−1/m

− m(1 + ϕ

−



−μ + μ

1−1/m  L ∞ (Ω) ) − e − K t )1−1/m

−

m(ηm + ϕ

 0 in Ω × (0, +∞), h = ηm on ∂Ω × [0, +∞) and h(·, 0) = ηm + ϕ −  g η,o in Ω . Thus the comparison principle gives that

g η (x, t )  h(x, t ) = ηm + ϕ − (x)e − K t

for (x, t ) ∈ Ω × [0, +∞),

where K > 0 depends only on the initial data go . Thus, we find c 0 > 0 such that

g η (x, t ) > ηm + c 0 dist(x, ∂Ω)e − K t ,

∀(x, t ) ∈ Ω × [0, +∞)

and |∇ g η (x, t )| > c 0 e − K t for (x, t ) ∈ ∂Ω × (0, +∞) since inf∂Ω |∇ ϕ − | > 0. On the other hand, let ϕ + be the positive eigenfunction of

⎧ ⎨ −M+  D 2 ϕ +  = ⎩

ϕ+ = 0

1 m−1



ϕ+

 m1

in Ω, on ∂Ω

from Theorem 3.4. Multiplying a positive constant to ϕ + , we assume that g η,o  ηm + ϕ + and that ϕ + is the positive eigenfunction associated with the eigenvalue μ+ > 0 since gη,o − ηm  2go and go ∈ Cb (Ω). If we define h := ηm + ϕ + , then h satisfies











mh1−1/m F D 2 h − ht  mh1−1/m M+ D 2 h = mh1−1/m −μ+



ϕ+

 m1

< 0 in Ω × (0, +∞).

From the comparison principle, we obtain that

g η  ηm + ϕ +

in Ω × [0, +∞).

Thus there is a uniform constant c 1 > 0, depending only on go , such that g η (x, t ) < ηm + c 1 dist(x, ∂Ω) for (x, t ) ∈ Ω × (0, +∞) and |∇ g η | < c 1 on ∂Ω × (0, +∞) since ϕ + ∈ C 0,1 (Ω). 2 Lemma 5.6. Under the same condition as Lemma 5.5, we also assume that g η,o converges to go in C γ (Ω) 1

(0 < γ  1) when η tends to 0. Let u be the solution of (5.16) with the initial data u o := gom and let g := um . Then for any T > 0, g η converges to g uniformly in Ω × [0, T ], up to a subsequence, when η tends to 0.

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

3291

Proof. Let 0 < ε < 1. From Lemma 5.5, we have

0 < δ  gη  M

in Ω(−ε) × [0, T ]

for δ := c 0 ε e − K T and M := c 1 diam(Ω), where Ω(−ε) := {x ∈ Ω : dist(x, Ω) > ε }. Thus g η satisfies



 ± M± ˜

˜Λ δλ, M





D 2 g η − ∂t g η  0 in Ω(−ε) × (0, T ]

˜ := mM 1−1/m . Then { g η } are equicontinuous in Ω(−2ε) × [0, T ] from [26] for δ˜ := mδ 1−1/m and M since g η,o ∈ C γ (Ω) for 0 < γ  1. We use Arzelà–Ascoli Theorem to deduce that g η converges to a continuous function locally uniformly in Ω × [0, T ], as η tends to 0, up to a subsequence. We recall that the family of viscosity solutions is closed in the topology of local uniform convergence, and that 0 < g η  ηm + c 1 dist(x, ∂Ω)

in Ω × [0, +∞)

from Lemma 5.5. Therefore, we deduce that g η converges to g uniformly in Ω × [0, T ], up to a subsequence, as η tends to 0, since g η,o converges to go in C γ (Ω) and the solution to (5.16) is unique. 2 m−1

m−1

From the above lemma, it suffices to show the concavity of g η2m = u η 2 cavity of the pressure of u.

for the square-root con-

Lemma 5.7 (Aronson–Bénilan inequality). Suppose that F satisfies ( F 1), ( F 3) and F (0) = 0. Let g η be the 1/m

solution of (5.23) with the initial data g η,o ∈ C 0 (Ω) satisfying (5.24) for some go ∈ Cb (Ω), and u η := g η Then we have

∂t g η  −

m m−1

·

gη t

and ∂t u η  −

1 m−1

·

uη t

,

∀(x, t ) ∈ Ω × (0, +∞).

.

(5.25)

Proof. (i) First, we assume that F is smooth. Let δ > 0 and let C be any positive constant bigger than m . We can select τδ ∈ (−δ, 0) such that m−1

∂t g η (x, δ) + C

g η (x, δ)

δ + τδ

 ηm ,

∀x ∈ Ω,

since ∂t g η (·, δ) is bounded and g η (·, δ)  ηm in Ω . We define

 g η (x, t ) . Z (t ) := inf ∂t g η (x, t ) + C x∈Ω t + τδ We note that Z (δ)  ηm > 0. Suppose that there is to ∈ (δ, +∞) such that Z (t o ) = 0. We may assume that t o is the first time for Z to vanish and hence Z t (to )  0. Since

∂t g η + C



gη t + τδ

=0+C

gη t + τδ

> 0 on ∂Ω × [δ, +∞),

the infimum of Z at time t = t o is achieved at an interior point xo of Ω . Then we have that g2

∂t g η (xo , to ) < 0, and (∂t g η )2 = C 2 (t +ητ )2 > 0 at the minimum point (xo , to ) ∈ Ω × (δ, +∞). o δ

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We consider the function 1−1/m

Ψ (s) := mg η

  (x, t ) F (1 − s) D 2 g η (x, t ) − (1 − s)∂t g η (x, t )

for any (x, t ) ∈ Ω × (0, +∞). We note that Ψ (0) = Ψ (1) = 0. We use the concavity of F to obtain that g η satisfies 1−1/m

mg η





F i j D 2 g η D i j g η − ∂t g η  0 in Ω × (0, +∞).

At the minimum point (xo , to ), we have

Z t = ∂t ∂t g η + C

= 1−

1 m



gη t + τδ −1/m

mg η

 (xo , to ) 



1−1/m

F D 2 g η ∂t g η + mg η





F i j D 2 g η · D i j ∂t g η + C

∂t g η t + τδ

−C





  gη 1 (∂t g η )2 1−1/m = 1− + mg η F i j D 2 g η · D i j ∂t g η + C m gη t + τδ

   ∂t g η gη gη 1−1/m +C − mg η F i j D 2 gη · D i j C −C t + τδ t + τδ (t + τδ )2 

∂t g η ∂t g η gη 1 (∂t g η )2  1− −C +C −C m gη t + τδ t + τδ (t + τδ )2 



gη gη gη 1 m−1 > 0, = 1− C2 − C = C − 1 C m m (t + τδ )2 (t + τδ )2 (t + τδ )2

gη

(t + τδ )2

which is a contradiction since Z t (to )  0. Therefore we deduce that Z (t ) > 0 for any t > δ , i.e.,

∂t g η > −C Since δ > 0 and C >

m m−1

gη t + τδ

 −C

gη t −δ

,

∀t > δ.

are arbitrary, we conclude that

∂t g η  −

m m−1

·

gη t

,

∀(x, t ) ∈ Ω × (0, +∞)

and hence the result follows. (ii) In general, we approximate F by smooth operators F ε using the mollification as in Lemma 4.7. Let g ηε be the solution of (5.23) with the operator F ε and the same initial data g η,o . Let g η± be the solution of (5.23) with the operator M± and the initial data g η,o , respectively. From the comparison principle, we have that

0 < ηm  g η−  g ηε  g η+ < +∞ in Ω × (0, +∞), and hence g ηε solves the uniformly parabolic equation in Ω × (0, +∞), where 0 <

η < 1 is fixed.

Then, g ηε and ∂t g ηε converge to g η and ∂t g η , locally uniformly in Ω × (0, +∞), respectively, when ε goes to 0, up to a subsequence. In fact, gηε converges to gη in C 1,α (Ω × (0, +∞)) since gηε and g η are uniformly bounded, where 0 < η < 1 is fixed. From the uniform interior C 2,α -estimate of the uniformly parabolic equation [26], we have the convergence of ∂t g ηε to ∂t g η locally uniformly in

Ω × (0, +∞) when ε tends to 0. Therefore, we use (i) for g ηε , to conclude that

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

∂t g η  − completing the proof.

m m−1

·

gη t

3293

for t > 0,

2

Corollary 5.8 (Aronson–Bénilan inequality). Suppose that F satisfies ( F 1), ( F 3) and F (0) = 0. Let u be the γ solution of (5.16) with um o ∈ Cb (Ω) ∩ C (Ω) for some 0 < γ  1. Then we have

∂t u  −

1 m−1

·

u t

in Ω × (0, +∞).

γ Proof. For 0 < η < 1, we find g η,o ∈ C γ (Ω) satisfying (5.24) and converging to go := um o in C (Ω) m as η goes to 0. Let g η be the solution of (5.23) with the initial data g η,o , and let g := u . From Lemma 5.6, g η converges to g uniformly in each compact subset of Ω × [0, +∞) when η tends to 0, up to a subsequence. As in the proof of Lemma 5.6, we have the local uniform Hölder estimate of g η in Ω × (0, +∞) and then we use Theorem 1.3 and Theorem 4.8 in [26] to obtain the uniform interior C 2,γ˜ -estimate of g η (0 < γ˜ < 1) in each compact subset of Ω × (0, +∞). Thus, ∂t g η converges to ∂t g locally uniformly in Ω × (0, +∞), when η goes to 0, up to a subsequence. Therefore, Lemma 5.7 and the convergence of g η to g imply that

∂t g  − which finishes the proof.

m m−1

·

g t

in Ω × (0, +∞),

2

Lemma 5.9. Suppose that F satisfies ( F 1), ( F 3) and F (0) = 0. Let g η be the solution of (5.23) with the initial data g η,o ∈ C 2,γ (Ω) (0 < γ < 1) satisfying (5.24) for some go ∈ Cb (Ω), and





F D 2 g η,o  0

in Ω.

Then g η is nonincreasing in time. Proof. Fix 0 < η < 1 and T > 0. We show that

∂t g η  0 in Ω × (0, T ]. We approximate the operator F by smooth operators F ε using mollification as in Lemma 4.7 and we may assume that F ε (0) = 0 by subtraction of F ε (0) to F ε . We also approximate the initial data g η,o by g ηε ,o satisfying F ε ( D 2 g ηε ,o ) = 0 on ∂Ω and F ε ( D 2 g ηε ,o )  0 in Ω . Indeed, let g ηε ,o be the solution of the following elliptic problem

√     ε 2  F D h = ξε · F ε D 2 g η,o − 2 nΛε in Ω, h = ηm

on ∂Ω,

where 0  ξε  1 satisfies that ξε ∈ C o∞ (Ω), and ξε ≡ 1 in Ω(−ε) for Ω(−ε) := {x ∈ Ω : dist(x, ∂Ω) > ε }. √ √ Then the solution g ηε ,o is such initial data since F ε ( D 2 g η,o )  F ( D 2 g η,o ) + 2 nΛε  2 nΛε as in the proof of Lemma 4.7. We notice that g ηε ,o and g η,o have a uniform C 1,γ -estimate in Ω for 0 < γ < 1 √ from [6] since F ( D 2 g η,o ) and ξε · { F ε ( D 2 g η,o ) − 2 nΛε } are uniformly bounded. Then Arzelà–Ascoli ε Theorem gives that g η,o converges to g η,o uniformly in Ω , as ε tends to 0, up to a subsequence, where 0 < η < 1 is fixed.

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Let g ηε be the solution of (5.23) with F ε and g ηε ,o in place of F and go , respectively. From the global Hölder regularity [26] and Arzelà–Ascoli Theorem, we have that g ηε converges uniformly to g η in Ω × [0, T ], as ε tends to 0, up to a subsequence. We recall that g ηε ∈ C 2,γ (Ω × [0, T ]) (see [26]) solves a uniformly parabolic equation (5.23) in Ω × (0, T ], where ε and η are fixed. For a fixed ε > 0, we will show that ∂t g ηε  0 in Ω × (0, T ]. Define

h := ∂t g ηε − δt − δ for small 0 < δ < 1. Then h is negative on the parabolic boundary of Ω × (0, T ]. Indeed, for (x, t ) ∈ ∂Ω × [0, +∞), we have that h = 0 − δt − δ  −δ < 0, and for (x, 0) ∈ Ω × {t = 0}, we have

h = ∂t g ηε − δ  −δ < 0 in Ω since ∂t g ηε (·, 0) = m( g ηε ,o )1−1/m F ε ( D 2 g ηε ,o )  0 in Ω . Suppose that there is to ∈ (0, T ] such that h(xo , to ) = 0 at some point xo ∈ Ω for the first time. Then the point (xo , to ) is a maximum point of h in Ω × (0, t o ], and hence at the maximum point (xo , to ), we have



0  m g ηε



ε 2 1−1/m ε  2 ε  1 (∂t g η ) F i j D g η D i j h − ht = − 1 − +δ ε m

gη

 



1 δ 2 (to + 1)2 1 δ 2 ( T + 1)2 + δ  − 1 − + δ. =− 1− ε m m

gη

m

η

However, it is a contradiction if we select δ small enough. Thus, for given 0 < ε , η < 1, and T > 0, we find a small δ(η, T ) > 0 such that if 0 < δ < δ(η, T ), then h < 0 in Ω × [0, T ], i.e.,

∂t g ηε < δt + δ in Ω × [0, T ]. Letting δ go to 0, we have ∂t g ηε  0 in Ω × [0, T ], i.e., for (x, t ) ∈ Ω × [0, T ],

g ηε (x, t + s) − g ηε (x, t )  0, Using the uniform convergence of g ηε to g η , we let

∀ s > 0.

ε go to 0 to deduce

∂t g η  0 in Ω × [0, T ], completing the proof.

2

Lemma 5.10. Suppose that a smooth operator F satisfies ( F 1), ( F 3) and F (0) = 0. Let g η be the solution of (5.23) with the initial data g η,o ∈ C 2,γ (Ω) (0 < γ < 1) satisfying (5.24) for some go ∈ Cb (Ω). We assume that F ( D 2 g η,o ) = 0 on ∂Ω and 1   −C˜ g ηm,o  F D 2 g η,o  0 in Ω

for some C˜ > 0. Then we have

−mC˜ g η  ∂t g η  0 in Ω × (0, +∞).

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

3295

Proof. According to Lemma 5.9, ∂t g η is nonpositive in Ω × (0, +∞). Define a linearized operator 1−1/m

H [h] := mg η

F i j · D i j h − ∂t h + 1 −

1



∂t g η

m

gη

h

for F i j := ∂∂pF ( D 2 g η ). Then we have that H [∂t g η ] = 0 in Ω × (0, +∞), and − g η satisfies ij

H [− g η ]  0 + 1 −

1 m



∂t g η gη

(− g η )  0 in Ω × (0, +∞)

since F is concave (see the proof of Lemma 5.7), and ∂t g η  0 in Ω × (0, +∞). We note that 1

1− −mC˜ g η,o  ∂t g η,o = mg η,o m F ( D 2 g η,o ) in Ω and that −mC˜ g η < 0 = ∂t g η on ∂Ω × [0, +∞). Therefore the result follows from the comparison principle. 2

Lemma 5.11. Suppose that a smooth operator F satisfies ( F 1), ( F 3) and F (0) = 0 and that Ω is strictly convex. Let g η be the solution of (5.23) with the initial data g η,o ∈ C 2,γ (Ω) (0 < γ < 1) satisfying (5.24) for some go ∈ Cb (Ω). We assume that F ( D 2 g η,o ) = 0 on ∂Ω , and

  1/m −C˜ g η,o  F D 2 g η,o  0 in Ω for a uniform constant C˜ > 0 with respect to 0 < η < 1. We also assume that g η,o has a uniform C 2 -estimate in Ω with respect to 0 < η < 1. Then for T > 0, we have

 2   D g η  < C ( T ) uniformly on ∂Ω × [0, T ], x

(5.26)

where C ( T ) > 0 depends only on m, n, λ, Λ, T , C˜ , the boundary gradient estimate of go , and the uniform C 2 -estimate of g η,o . Proof. (i) Since g η,o has a uniform Lipschitz estimate with respect to 0 < Lemma 5.9 imply that

η < 1, Lemma 5.5 and

|∇ g η | < C uniformly in Ω × [0, +∞),

(5.27)

where C > 0 is uniform with respect to 0 < η < 1. In fact, for any unit vector e α ∈ Rn , ∂α g η satisfies 1−1/m

mg η

F i j · D i j (∂α g η ) − ∂t (∂α g η ) + 1 −

1



m

∂t g η gη

∂α g η = 0 in Ω × (0, +∞)

for F i j := ∂∂pF ( D 2 g η ). Since ∂t g η  0 from Lemma 5.9, the comparison principle implies that ∂α g η is ij uniformly bounded by a constant depending only on the uniform Lipschitz estimate of g η,o , and the uniform boundary gradient estimate in Lemma 5.5. (ii) We fix η > 0, and denote g η by g for simplicity. We fix a boundary point xo ∈ ∂Ω and denote xo by the origin. Now we introduce the coordinate system such that the tangent plane to ∂Ω at 0 is xn = 0 with en being the inner normal vector. When e τ = e i (i = 1, . . . , n − 1) is tangential to ∂Ω at 0, we have g τ = 0 and g τ τ = − g ,n κτ < 0 at 0 as in the proof of Lemma 4.9, where κτ is the curvature of ∂Ω at 0 in the direction e τ . Using the uniform boundary estimates in Lemma 5.5, and the strict convexity of ∂Ω , we obtain

0 < c ( T ) < − g τ τ (0, t ) < C ,

∀t ∈ [0, T ]

(5.28)

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S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

for any tangential unit vector e τ to ∂Ω at 0, where 0 < c ( T ) < C are independent of 0 < η < 1. Thus, there is C > 0, independent of 0 < η < 1, such that

  ∂e ,e g (0, t ) < C , i j

∀t ∈ [0, T ] (1  i , j  n − 1).

(iii) Near the origin, ∂Ω is represented by xn = ψ(x ) = 12 A i j xi x j + O (|x |3 ). Since Ω is strictly convex, the eigenvalues of ( A i j ) lie in [κ0 , κ1 ] for some 0 < κ0 < κ1 . After a change of coordinate of Rn−1 , ˜ x ) = 1 |x |2 + O (|x |3 ) and the operator F will be transthe boundary of Ω near 0 becomes xn = ψ( 2

˜ = λ˜ (λ, Λ, κ0 , κ1 ) and Λ˜ = Λ(λ, ˜ Λ, κ0 , κ1 ) formed to a new operator F˜ with new elliptic coefficients λ that are uniformly bounded and positive. So ∂Ω is close to a unit ball with an error O (|x |3 ) near the origin. For simplicity, we assume that Ω = B 1 (en ) and denote B 1 (en ) by B 1 . The general domain can be considered with a simple modification as [8]. (iv) We claim that |∂ek ,en g (0, t )|  C for t ∈ [0, T ], where C > 0 is independent of η , and k = 1, . . . , n − 1. For positive constants A 1 , A 2 , and A 3 , which will be fixed later, we define w ± (x, t ) := ∂ T k g ± A 1

n −1 



g ,2l ± A 2 xn2 ± A 3 1 − |x − en |2

2 − ρ

l =1

:= (1 − xn ) g ,k + xk g ,n ± A 1

n −1 



g ,2l ± A 2 xn2 ± A 3 1 − |x − en |2

2 − ρ

,

l =1 1 where ρ := 1 − m , and ∂ T k g := (1 − xn ) g ,k + xk g ,n is a directional derivative and coincides with a tangential derivative on ∂ B 1 . Define

v := A 4 xn for a uniform constant A 4 > 0, which will be chosen large later. Now, we consider a linearized operator 1

H [h] := mg 1− m F i j · D i j h − ∂t h with F i j := ∂∂pF ( D 2 g ). We will show that ij

H [w +]  0 = H [v ]

and

H [ w − ]  0 = H [− v ]

in B 1 × (0, T ]

(5.29)

for sufficiently large constants A 1 , A 2 , and A 3 . We can select A 4 > 0 large so that

− v  w −  w +  v on B 1 × {0} since g η,o has a uniform C 2 -estimate in Ω , and satisfies that ∂ T k g η,o = 0 on ∂ B 1 , ∂l g η,o (0) = 0 for l = 1, . . . , n − 1, and |x|2  2xn for x ∈ B 1 . We recall that |∇x g | < c 1 on ∂ B 1 × [0, +∞) from Lemma 5.5. Since g = ηm on ∂ B 1 and



g ,2l = (1 − xn ) g ,l + xl g ,n + xn g ,l − xl g ,n

2

 2  2 (1 − xn ) g ,l + xl g ,n + 2(xn g ,l − xl g ,n )2

 2  2 (1 − xn ) g ,l + xl g ,n + 8c 12 |x|2 = 8c 12 |x|2 on ∂ B 1 × [0, +∞),

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

3297

we see that for (x, t ) ∈ ∂ B 1 × [0, +∞),

  − 8(n − 1)c 12 A 1 + A 2 |x|2  w − (x, t )  w + (x, t )  8(n − 1)c 12 A 1 + A 2 |x|2 . Since |x|2 = 2xn for x ∈ ∂ B 1 , we obtain that

  −2 8(n − 1)c 12 A 1 + A 2 xn  w −  w +  2 8(n − 1)c 12 A 1 + A 2 xn ,

∀(x, t ) ∈ ∂ B 1 × [0, T ].

Thus for a large A 4 > 0, we have that

  − v  w −  w +  v on ∂ p B 1 × (0, T ] . If we prove (5.29), then the comparison principle gives

− v  w −  w +  v in B 1 × [0, T ]. Therefore, we deduce that, for 1  k  n − 1,

      ∂kn g (0, t ) = ∂n w ± (0, t )  ∂n v (0) = A 4 for t ∈ [0, T ]. So, it remains to show (5.29) by choosing suitable constants A 1 , A 2 , and A 3 . Note that ∂t g η / g η is uniformly bounded in Ω × (0, +∞) from Lemma 5.10. We use (5.27), Lemma 5.10, and the ellipticity of F to have

  

n −1  1 ∂t g 2 H [w+]  − 1 − g ,l (1 − xn ) g ,k + xk g ,n + 2 A 1 m

g

 1

+ 2mg 1− m −

l =1 n 

F ni g ,ki +

i =1

n 

F ki g ,ni + A 1

i =1

n −1 

F i j g ,li g ,lj + A 2 F nn

l =1

− ρ 1  4λ|x − en |2 + A 3 (2 − ρ )(1 − ρ )mg 1− m 1 − |x − en |2 1 − ρ 1  − A 3 (2 − ρ )mg 1− m 1 − |x − en |2 nΛ   −C (1 + A 1 ) + 2mg

1 1− m

 −C

n  l , i =1

1 2

| g ,li |

2

+ A1λ

n −1  n 

 g ,2li

+ A 2 λ/2

l =1 i =1

− ρ 1 1  + mg 1− m A 2 λ + A 3 (2 − ρ )(1 − ρ )mg 1− m 1 − |x − en |2 4λ|x − en |2 1 − ρ 1  − A 3 (2 − ρ )mg 1− m 1 − |x − en |2 nΛ 1

for a uniform C > 0 with respect to 0 < η < 1. Using the equation mg 1− m F ( D 2 g ) − ∂t g = 0 = F (0) and the ellipticity of F , it follows that (see the proof of Theorem 9.5 in [6])

g ,2nn

C

 (i , j )=(n,n)

   ∂t g 1/m 2  | g ,i j | + C  · g  in B 1 × (0, +∞). g

Using (5.30) and Lemma 5.10, we have

2

(5.30)

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S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

 H [ w + ]  −C (1 + A 1 ) + 2mg

1 1− m



n 

−C

1 2

| g ,li |

2

+

n A1 

C

l , i =1

 g ,2li

+ A 2 λ/2

l , i =1

− ρ 1 1  4λ|x − en |2 + mλ A 2 g 1− m + A 3 (2 − ρ )(1 − ρ )mg 1− m 1 − |x − en |2 1 − ρ 1  − A 3 (2 − ρ )mg 1− m 1 − |x − en |2 nΛ 3

for a large C > 0, independent of 0 < η < 1. Selecting A 2  A 1 and A 21  C2λ (see [6, Theorem 9.5]), we obtain 1

1



H [ w + ]  −C (1 + A 1 ) + mλ A 2 g 1− m + A 3 (2 − ρ )(1 − ρ )mg 1− m 1 − |x − en |2

− ρ

4λ|x − en |2

1/m 1  − A 3 (2 − ρ )mg 1− m 1 − |x − en |2 nΛ.

Since g η  δ(1 − |x − en |2 ) in B 1 × (0, T ] for a uniform δ = δ( T ) > 0 with respect to 0 < η < 1 from Lemma 5.5, we have 1



H [ w + ]  −C (1 + A 1 ) + mλ A 2 δ 1− m 1 − |x − en |2

1− m1

  1/m 1 + A 3 m(2 − ρ )(1 − ρ )δ 1− m 4λ|x − en |2 − C 1 − |x − en |2

for a large C > 0 independent of 0 < η < 1. Choosing A 3 and A 2 large, we deduce H [ w + ]  0 in B 1 × (0, T ], and hence

H [w +]  0 = H [v ]

in B 1 × (0, T ].

Similarly, we have H [ w − ]  0 = H [− v ] in B 1 × (0, T ]. Therefore, we have proved that

  ∂kn g (0, t )  A 4 for all t ∈ [0, T ], where A 4 is uniform with respect to 0 < η < 1. (v) Lastly, since g ,2nn (0, t )  C (i , j )=(n,n) | g ,i j (0, t )|2 for (0, t ) ∈ ∂Ω × (0, +∞) from (5.30), we have

  ∂nn g η (0, t )  C ,

∀t ∈ (0, T ],

where C > 0 is independent of 0 < η < 1. Therefore, we have

 2   D g η (0, t )  C , x

∀t ∈ [0, T ],

and hence (5.26) follows since xo = 0 is an arbitrary point of ∂Ω .

2

Lemma 5.12. Suppose that F satisfies ( F 1), ( F 3) and F (0) = 0 and that Ω is strictly convex. Let g η be the solution of (5.23) with the initial data g η,o ∈ C 2,γ (Ω) (0 < γ < 1) satisfying (5.24) for some go ∈ Cb (Ω). We also assume that for T > 0,

 2   D g η  < C ( T ) uniformly on ∂Ω × (0, T ], x m−1

(5.31)

where C ( T ) > 0 is independent of 0 < η < 1. Let w η := g η2m . Then, there exist η( T ) > 0 and c ( T ) > 0 such that if 0 < η < η( T ), then

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306



m−1

w η,αα (x, t ) =

−1 2− m2m

g η g η,αα −

m+1

2mg η

−

m − 1 c(T ) 2m

η

m +1 2

2m

3299

 g η2 ,α

∀(x, t ) ∈ ∂Ω × (0, T ],

,

(5.32)

for any unit vector e α ∈ Rn , where η( T ) > 0 and c ( T ) > 0 depend only on C ( T ), c 0 e − K T and the lower bound of the curvature of ∂Ω , and the uniform constant c 0 e − K T > 0 is as in Lemma 5.5. Proof. We fix η > 0. For simplicity, we denote g η by g. Fix an (xo , to ) ∈ ∂Ω × (0, T ]. We may assume xo = 0 and introduce the coordinate system such that xo = 0 and that the tangent plane at 0 is xn = 0 with en being the inner normal vector at the origin. When e τ = e i (i = 1, . . . , n − 1) is tangential to ∂Ω at xo = 0, we have g τ = 0 and g τ τ = − g ,n κτ < 0 at 0 as in the proof of Lemma 4.9, where κτ > 0 is the curvature of ∂Ω at 0 in the direction e τ . According to the uniform boundary gradient estimates in Lemma 5.5 and the strict convexity of ∂Ω , we have

0 < c 1 ( T ) < − g τ τ = |∇ g | · κτ < C 1

at (0, to ) ∈ ∂Ω × (0, T ]

(5.33)

for any tangential unit vector e τ to ∂Ω at 0, where c 1 ( T ) > 0 and C 1 > 0 are uniform with respect to 0 < η < 1. Then we have

g (0, to ) · g τ τ (0, to ) −

m+1 2m

g τ2 (0, to )  −c 1 ( T )ηm − 0 = −c 1 ( T )ηm

for any tangent unit vector e τ to ∂Ω at xo = 0 ∈ ∂Ω , where c 1 ( T ) > 0 depends only on c 0 e − K T and the lower bound of the curvature of ∂Ω . Let e α be any unit vector in Rn . We decompose e α := β1 e τ + β2 e ν with β12 + β22 = 1, where unit vectors e ν and e τ are normal and tangent to ∂Ω at 0, respectively. We use (5.33), (5.31) and Lemma 5.5 to have at (0, t o ) ∈ ∂Ω × (0, T ],

gg ,αα (0, to ) −

m+1 2m





g ,2α (0, to ) = g β12 g τ τ + 2β1 β2 g τ ν + β22 g νν −

m+1 2m

β22 g ν2

    g −c 1 ( T )β12 + C ( T ) 2β1 β2 + β22 − β22 δo ( T )

for a uniform δo ( T ) > 0 depending on m, c 0 e − K T . We use Young’s inequality to deduce that at (0, t o ) ∈ ∂Ω × (0, T ],

gg ,αα (0, to ) −

m+1 2m

  c1 (T ) 2 2 ˜ β1 + C ( T )β2 − β22 δo ( T ) g ,α (0, to )  g − 2

2

= −ηm − with C˜ ( T ) := C ( T ) +

2C ( T )2 , c1 (T )

c1 (T ) 2

 β12 + ηm C˜ ( T ) − δo ( T ) β22

min{c 1 ( T ), δo ( T )} 2





ηm β12 + β22 =: −c ( T )ηm

δ (T )

for small 0 < ηm < {η( T )}m := o˜ . Thus we conclude 2C ( T )

g η (xo , to ) g η,αα (xo , to ) −

m+1 2m

g η2 ,α (xo , to )  −c ( T )ηm

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for any direction e α . Since (xo , to ) is an arbitrary point of ∂Ω × (0, T ], we deduce

w η,αα =

m−1 2m

η

completing the proof.



1 1 2m− m− 2

g η g η,αα −

m+1 2m

 g η2 ,α

−

m − 1 c(T ) 2m

η

m +1 2

on ∂Ω × (0, T ],

2

Remark 5.13. To obtain the uniform estimate (5.31) up to the boundary, we need to prove a uniform 1/m 2, α up to the boundary, which will be studied in the future work. weighted C δ -estimate of u η = g η When F ( D 2 u ) = u, Schauder theory has been proved in [16].

Lemma 5.14. Suppose that F satisfies ( F 1), ( F 3) and F (0) = 0 and that Ω is strictly convex. Then there exist go ∈ Cb (Ω) ∩ C 2,γ (Ω) (0 < γ < 1), and 0 < ηo < 1 satisfying the following properties: (i) go and g η,o := go + ηm satisfy





F D2ψ = 0

on ∂Ω,

and

  −C˜ ψ 1/m  F D 2 ψ  0 in Ω

for a uniform constant C˜ > 0 with respect to 0 < η < ηo and F , (ii) go and g η,o have a uniform C 2,γ -estimate in Ω with respect to 0 < η < ηo and F , m −1

m−1

(iii) go 2m and g η2m ,o are concave in Ω for 0 < η < ηo , (iv) g η,o converges to go in C 2 (Ω) as η tends to 0, where ηo > 0 is a uniform constant. Proof. Let d be the distance function to ∂Ω , which is concave in Ω , and let h be the solution to

 



1

F D 2 h = −d m

in Ω,

h=0

on ∂Ω.

Note that h has a uniform C 2,γ -estimate (0 < γ < 1) in Ω with respect to F . Then there exist uniform numbers εo , ηo > 0, and a convex domain Ωo  Ω such that |∇ h| > εo in Ω \ Ωo from Hopf ’s Lemma, the level sets of h are strictly convex in Ω \ Ωo , and that D 2 (h + ηm ) since h ∈ C 2,γ (Ω), and

D

2



h+η

m

−1  m2m

m−1 2m

 0 in Ω \ Ωo for 0  η < ηo



 m −1 −2   m − 1 m+1 = h + ηm 2m h + ηm D 2 h − ∇ h ∇ ht 2m 2m



(we refer to (4.14), and the proof of Lemma 5.12). Let δ0 < δ1 < δ2 be small positive numbers to be fixed later satisfying Ω \ {h > δ2 } ⊂ Ω \ Ωo . Define

h˜ := h − C o (h − δ0 )3

in {h > δ0 } \ Ωo

for some C o > 0, which will be chosen later. Then we can find 0 < δ0 < δ1 < δ2 < 1, and C o > 0 such that ∇ h˜ ∇ h, 12 < ∇ h˜ · ∇ h < 1 in {h > δ0 } \ {h > δ2 }, D 2 h˜  0 in {h > δ1 } \ {h > δ2 }, and that F ( D 2 h˜ )  0, D 2 (h˜ + ηm ) Now we set

m−1 2m

go := h

 0 in {h > δ0 } \ {h > δ1 } for 0  η < ηo .

in Ω \ {h > δ0 },

go := h˜

in {h > δ0 } \ {h > δ2 },

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

3301

and

go := δ2 − C o (δ2 − δ0 )3

on {h > δ2 }.

Then go is concave in {h > δ1 }. By regularizing go in {h > (δ1 + δ2 )/2}, we obtain go ∈ C 2,γ (Ω) such that go is concave in {h > δ1 }, and that



1   −C˜ gom  F D 2 go  0 in Ω \ {h > δ1 },

go = 0

on ∂Ω m−1

for some C˜ > 0 since h ∈ Cb (Ω) ∩ C 2,γ (Ω), and F ( D 2 h˜ )  0 in {h > δ0 } \ {h > δ1 }. We notice that go 2m is concave in {h > δ1 } since go is concave in {h > δ1 }. Therefore, we set g η,o := go + ηm , and hence go and g η,o satisfy (i)–(iv), where we recall D 2 ( go + ηm )

m−1 2m

 0 in Ω \ {h > δ1 } for 0  η < ηo . 2

Lemma 5.15. Suppose that F satisfies ( F 1), ( F 2) and ( F 3) and Ω is a strictly convex bounded domain. Let 1

go be an initial data in Lemma 5.14, u be the solution of (5.16) with initial data u o := gom , and g := um . Then u

m−1 2

is concave in the spatial variables for any t > 0, i.e.,

D 2x u

m −1 2

 0 in Ω × (0, +∞).

Proof. Let g η be the solution of (5.23) with the initial data g η,o , where g η,o is as in Lemma 5.14. For m−1

m−1

a fixed T > 0, it suffices to show the concavity of w η := u η 2 = g η2m for small 0 < η  1;

1 2

m −1

m −1



m −1



g η2m (x, t ) + g η2m ( y , t ) − g η2m

x+ y 2

 , t  0,

∀x, y ∈ Ω, ∀t ∈ [0, T ]

since the uniform convergence of g η to g in Lemma 5.6 preserves the concavity. Now, we approximate F by a smooth operator F ε as in Lemma 4.7, and we may assume that ε F (0) = 0. For any ε > 0, let goε and g ηε ,o be the initial data as in Lemma 5.14 with F ε in place of F , and let g ηε be the solution of (5.23) with F ε and g ηε ,o . We note that Lemmas 5.11 and 5.12 hold for g ηε for 0 < ε < 1 and 0 < η < min{ηo , η( T )}, where η( T ) and ηo are the uniform constants as in Lemma 5.12 and Lemma 5.14, respectively. Fix 0 < η < min{η( T ), ηo }. Since g ηε ,o is uniformly bounded with respect to 0 < ε < 1, where η is fixed, the comparison principle implies

0 < ηm  g ηε  C < +∞ in Ω × (0, +∞). Since g ηε is uniformly bounded with respect to 0 < ε < 1, we have the uniform global C 1,γ -estimate (0 < γ < 1) for g ηε in Ω × [0, T ] with respect to 0 < ε < 1 from [26] and hence the uniform C 2,γ estimate for g ηε in Ω × [0, T ] using Theorem 1.1 in [26], where 0 < η < min{η( T ), ηo } is fixed. According to Arzelà–Ascoli Theorem, g ηε converges uniformly to g η in C 2 (Ω × [0, T ]), up to a subsequence, since g ηε ,o converges to g η,o uniformly in Ω as ε tends to 0, up to a subsequence. Thus we consider the concavity of w εη := (u εη )

m−1 2

for small 0 < ε < 1. We note that the function g ηε solves





mg 1−1/m F ε D 2 g = ∂t g which is uniformly parabolic for a given

η > 0.

in Ω × (0, T ],

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The geometric quantity w εη satisfies

∂t w =

m−1 2



m −3

w m −1 F ε

2m

3−m

m−1

w2 D2 w +

w m −1

m+1 m−1

 w D w D wt

in Ω × (0, T ].

After the change of the time t → mt, the above equation will be transformed to

∂t w = with r =

m+1 . m−1

∂t w ,α β =

m−1 2m

w

m −3 m −1

Fε

2m m−1

w

3−m m −1



2

2

w D w + rwDwDw

t



 ,

(5.34)

By taking differentiation twice, the function w εη satisfies

m−1 2m

·



m −3

w m−1 F iεj ,kl ·

2m

3−m

m−1

w m −1



2m m−1

3−m

w m −1

w 2 D kl w +

w2 Dij w +

m+1 m−1

m+1 m−1

 w Di w D j w ,α



w Dk w Dl w ,β

 + F iεj · 2w ,α w ,β D i j w + 2w w ,α β D i j w + 2w w ,α D i j w ,β + 2w w ,β D i j w ,α + w 2 D i j w ,α β + r w ,α β D i w D j w + 2r w ,α D i w ,β D j w + 2r w ,β D i w ,α D j w  + 2r w D i w ,α D j w ,β + 2r w D i w ,α β D j w 

 3−m  m − 1 m − 3 m −3 −1 2m + w m−1 w ,α β F ε w m −1 w 2 D 2 w + r w D w D w T 2m m − 1 m−1   m − 3 −1 − w w ,α β F iεj · w 2 D i j w + r w D i w D j w m−1 

 m −3 3−m  m−1m−3 2 2m − w m−1 −2 w ,α w ,β F ε w m −1 w 2 D 2 w + r w D w D w T 2m m − 1 m − 1 m−1   m−3 2 + w −2 w ,α w ,β F iεj · w 2 D i j w + r w D i w D j w , m−1m−1 3−m

ε

ε m−2 (( w ε )2 D 2 w ε + r w ε D w ε ( D w ε )t )) and with F iεj = ∂∂ pF ( m2m η η η η η −1 ( w η ) ij

F iεj ,kl =



3−m  2  t  ∂2 F ε 2m  ε  m . w η −2 w εη D 2 w εη + r w εη D w εη D w εη ∂ p i j ∂ pkl m − 1

In order to study the concavity of w εη , we consider for given 0 < ε < δ < 1,

sup sup ∂ββ w εη ( y , t ) + ψ(t ), y ∈Ω |e β |=1

√

where e β ∈ Rn is a unit vector, and ψ(t ) := −ε − e −1/δ e 2K t tan( K δt ) for some uniform constant K > 0, which will be chosen later and independent of 0 < ε < δ < 1. Now suppose that there is a time t o ∈ [0, min( π√ , T )] ⊂ [0, T ] such that 4K

δ

sup sup ∂ββ w εη ( y , to ) + ψ(to ) = 0.

y ∈Ω |e β |=1

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

3303

We may assume that t o is the first time for it to be zero. From the assumption on the initial data g ηε ,o that D 2 ( g ηε ,o )

m−1 2m

 0 in Ω , we have sup sup ∂ββ w εη ( y , 0) + ψ(0)  ψ(0) = −ε < 0,

y ∈Ω |e β |=1

and hence to > 0. We assume that the supremum

sup sup ∂ββ w εη (x, to ) = ∂αα w εη (xo , to ) = −ψ(to ) > 0

y ∈Ω |e β |=1

is achieved at (xo , to ) ∈ Ω × (0, min( π√ , T )] with some direction e α . From the boundary estimate 4K

δ

of the second derivatives, (5.32) in Lemma 5.12, the maximum point xo should be an interior point of Ω since ∂αα w εη (xo , to ) = −ψ(to ) > 0. Without losing of generality, we assume that xo = 0 and introduce orthonormal coordinates in which e α is taken as one of the coordinate axes so that

∂α β w εη (0, to ) = 0 for β = α . In order to create extra terms, we perturb second derivatives of w εη and we use the function

Z (x, t ) := ∂α β w εη (x, t )ξ α (x)ξ β (x) where ξ β (x) := δα β + c α xβ + 12 c α c α xα xβ and we denote ξt := (ξ 1 , . . . , ξ n ) (see [23]). We choose c β ∈ R so that

 2 −4 w εη c β + 4w εη ∂β w εη = 0 at the maximum point (0, to ). We notice that Z (0, t o ) = ∂αα w εη (0, to ) = −ψ(to ) > 0. Now we define









2 Y (x, t ) := Z (x, t ) + ψ(t ) ξ (x) = ξt (x) D 2x w εη (x, t ) + ψ(t )I ξ (x).

We have that

∂t Y  0,

D 2x Y  0 and

∇x Y = 0 at (0, to ),

since

D 2x w εη (x, t ) + ψ(t )I  0 for (x, t ) ∈ Ω × (0, to ],

and

 Y (0, to ) = ξt (0) D 2x w εη (0, to ) + ψ(to )I ξ (0) = ∂αα w εη (0, to ) + ψ(to ) = 0. A simple computation gives that at (0, t o ),

Z ,i = ∂α β i w εη ξ α ξ β + 2∂β i w εη c α ξ α ξ β , Z ,i j = ∂α β i j w εη ξ α ξ β + 4∂β i j w εη c α ξ α ξ β + 2∂β i w εη c j c α ξ α ξ β + 2∂i j w εη c α c β ξ α ξ β . Thus at the maximum point (0, t o ) of all second derivatives of w εη , we have

(5.35)

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S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

Z t = ∂α β t w εη ξ α ξ β

     w 2 F iεj · Z ,i j + F iεj · w ,β i j ξ α ξ β −4w 2 c α + 4w w ,α + 2F iεj · w ,i j − w 2 c α2 + w 2,α   + 2w F iεj · D i j w + r F iεj · D i w D j w w ,αα + 4r F αε j · D j w w ,α w ,αα ε · w2 − 2w 2 F αε j · c j c α w ,αα + 2r w F iεj · D j w D i w ,αα + 2r w F αα ,αα   √ |m − 3| − m+1 2 + w m−1 w w ,αα + w 2 2 n Λε 2m m − 1 ,α       w 2 F iεj · ∂i j |ξ |2 −ψ(to ) + 2w · nΛ∂αα w + r Λ|∇ w |2 Z

  + 4r Λ|∇ w |2 Z + 2w 2 Λ c 2j Z j

+ 2r w Λ Z 2 +

|m − 3| 2m

m +1



w − m−1 w w ,αα +



2 m−1

√

w 2,α 2 nΛε ,

where we use the concavity of F ε , (4.11) in Lemma 4.7, (5.35), ( F 1 ), and the choice of c α . n 2 1/ 2 Since F iεj · ∂i j (|ξ |2 ) = 2c α ( F c + F c )  2 ( n + 1 )Λ| c |( from ( F 1), w := w εη satj j j α j α α j cj) j =1 isfies that at (0, t o ),

Z t (0, to ) = ∂α β t w εη ξ α ξ β

  2w εη nΛ + 2r w εη Λ Z 2 

 

  2   2  2 + w εη · 2(n + 1)Λ c 2j + 5r Λ∇ w εη  + 2 w εη Λ c 2j j



|m − 3|  ε − m−2 1 √ wη 2 nΛ Z + +



2m

Since c β =

j

 2 √ |m − 3|  ε − mm+−11  ε ∂α w η · 2 n Λ ε . wη m(m − 1)

∂β w εη (0,to ) , we use the uniform global C 1,γ -estimate for g ηε (with respect to 0 w εη (0,to )

< ε < 1) to

find a large constant K = K η > 0 independent of 0 < ε < δ < 1 such that





Z t (0, to )  K Z 2 + Z + ε , where

η is fixed. Thus we obtain that 0  ∂t Y (0, to ) = Z t (0, to ) + ψt (to )

     K Z 2 + Z + ε + ψt (to ) = ψt (to ) + K ψ 2 (to ) − ψ(to ) + ε .

On the other hand, we can check that ψ(t ) := −ε − e −1/δ e 2K t tan( K

  ψt + K ψ 2 − ψ + ε < 0,

∀0 < t 

√

δt ) satisfies

π 4K

√ , δ

for small 0 < ε  δ  1, which are uniform numbers with respect to T , η, and K . This implies a contradiction to the fact that ∂t Y (0, to )  0 if to ∈ (0, min( π√ , T )]. Therefore, we deduce that for small 0 < ε  δ  1,

4K

δ

S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

sup sup ∂ββ w εη ( y , t ) < −ψ(t ), y ∈Ω |e β |=1

 ∀t ∈ 0, min

3305



π

√ ,T 4K δ

,

i.e.,



sup sup ∂ββ u εη

 m−2 1

y ∈Ω |e β |=1

( y , t ) < ε + e −1/δ+π /(2

Using the uniform C 2,γ -estimate of g ηε , we let δ and

√ δ)

,

 ∀t ∈ 0, min



π

√ ,T 4K δ

.

ε go to 0 to conclude that

m −1

D 2x u η 2  0 in Ω × (0, T ], which means

1 2

m −1



m −1

m −1



u η 2 (x, t ) + u η 2 ( y , t ) − u η 2

x+ y 2

 , t  0,

∀x, y ∈ Ω, ∀t ∈ [0, T ].

2

This finishes the proof.

Corollary 5.16 (Square-root concavity). Let F satisfy ( F 1), ( F 2) and ( F 3). If Ω is strictly convex, then φ 1 . concave, where φ is the positive eigenfunction Theorem 3.4 and p := m

1− p 2

is

Proof. We choose the initial data go as in Lemma 5.14. Let u be the solution of (5.16) with initial 1

data u o := gom . Lemma 5.15 implies

D 2x u

m −1 2

 0 in Ω × (0, ∞).

The uniform convergence in Corollary 5.3, namely, m

t m−1 um (x, t ) → φ(x)

uniformly for x ∈ Ω as t → +∞,

preserves the concavity. Therefore, it follows that

1 2

φ

m −1 2m

(x) + φ

m −1 2m

 m −1 x+ y  0 for x, y ∈ Ω. ( y ) − φ 2m 2

Corollary 5.17. Let F satisfy ( F 1), ( F 2) and ( F 3). If Ω is convex, then φ 1 eigenfunction Theorem 3.4 and p := m .

1− p 2

2

is concave, where φ is the positive

Acknowledgment Ki-Ahm Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2010-0001985).

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S. Kim, K.-A. Lee / J. Differential Equations 254 (2013) 3259–3306

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